SMITH NUMBERS

## Some Interesting Observations

----------------------------------------------------------------------------------------

Introduction

A composite integer N whose digit sum S(N) is equal to the sum of the digits of its prime factors Sp (N) is called a Smith number [17].

For example 85 is a Smith number because digit sum of 85 i.e. S(85) = 8 + 5=13, which is equal to the sum of the digits of its prime factors i.e. Sp (85) = Sp (17 x 5) = 1 + 7 + 5 = 13.

Albert Wilansky named Smith numbers from his brother-in-law Harold Smiths telephone number 4937775 with this property i.e.

4937775 = 3.5.5.65837

Since

4+9+3+7+7+7+5=3+5+5+(6+5+8+3+7)=42

Wilansky also mentioned two other numbers with this property i.e. 9985 and 6036.Wilansky has found that there are 360 Smith numbers less than 10000, which is not correct, as there are 376 Smith numbers less then 10000. It is now known that there are infinitely many Smith numbers [2].

There are 25154060 smith numbers below 109[4]. Further computations reveal that there are 241882509 smith numbers below 1010. Jens Kruse Andersen reported on 28th April 2008 that there are 2335807857 smith numbers below 1011.

All 376 Smith numbers below 10000 are :

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219, 1255, 1282, 1284, 1376, 1449, 1507, 1581, 1626, 1633, 1642, 1678, 1736, 1755, 1776, 1795, 1822, 1842, 1858, 1872, 1881, 1894, 1903, 1908, 1921, 1935, 1952, 1962, 1966, 2038, 2067, 2079, 2155, 2173, 2182, 2218, 2227, 2265, 2286, 2326, 2362, 2366, 2373, 2409, 2434, 2461, 2475, 2484, 2515, 2556, 2576, 2578, 2583, 2605, 2614, 2679, 2688, 2722, 2745, 2751, 2785, 2839, 2888, 2902, 2911, 2934, 2944, 2958, 2964, 2965, 2970, 2974, 3046, 3091, 3138, 3168, 3174, 3226, 3246, 3258, 3294, 3345, 3366, 3390, 3442, 3505, 3564, 3595, 3615, 3622, 3649, 3663, 3690, 3694, 3802, 3852, 3864, 3865, 3930, 3946, 3973, 4054, 4126, 4162, 4173, 4185, 4189, 4191, 4198, 4209, 4279, 4306, 4369, 4414, 4428, 4464, 4472, 4557, 4592, 4594, 4702, 4743, 4765, 4788, 4794, 4832, 4855, 4880, 4918, 4954, 4959, 4960, 4974, 4981, 5062, 5071, 5088, 5098, 5172, 5242, 5248, 5253, 5269, 5298, 5305, 5386, 5388, 5397, 5422, 5458, 5485, 5526, 5539, 5602, 5638, 5642, 5674, 5772, 5818, 5854, 5874, 5915, 5926, 5935, 5936, 5946, 5998, 6036, 6054, 6084, 6096, 6115, 6171, 6178, 6187, 6188, 6252, 6259, 6295, 6315, 6344, 6385, 6439, 6457, 6502, 6531, 6567, 6583, 6585, 6603, 6684, 6693, 6702, 6718, 6760, 6816, 6835, 6855, 6880, 6934, 6981, 7026, 7051, 7062, 7068, 7078, 7089, 7119, 7136, 7186, 7195, 7227, 7249, 7287, 7339, 7402, 7438, 7447, 7465, 7503, 7627, 7674, 7683, 7695, 7712, 7726, 7762, 7764, 7782, 7784, 7809, 7824, 7834, 7915, 7952, 7978, 8005, 8014, 8023, 8073, 8077, 8095, 8149, 8154, 8158, 8185, 8196, 8253, 8257, 8277, 8307, 8347, 8372, 8412, 8421, 8466, 8518, 8545, 8568, 8628, 8653, 8680, 8736, 8754, 8766, 8790, 8792, 8851, 8864, 8874, 8883, 8901, 8914, 9015, 9031, 9036, 9094, 9166, 9184, 9193, 9229, 9274, 9276, 9285, 9294, 9296, 9301, 9330, 9346, 9355, 9382, 9386, 9387, 9396, 9414, 9427, 9483, 9522, 9535, 9571, 9598, 9633, 9634, 9639, 9648, 9657, 9684, 9708, 9717, 9735, 9742, 9760, 9778, 9840, 9843, 9849, 9861, 9880, 9895, 9924, 9942, 9968, 9975, 9985.

Note that the Beast number 666 is also a Smith Number.

Various Kinds of Smith Numbers

• Smith Semiprimes:

The smith numbers which are product of two prime numbers can be termed as Smith Semiprimes . For example 22 is a Smith Semiprime (Sloane's A098837).

All Smith Semiprimes below 104 are:

4, 22, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 382, 391, 454, 517, 526, 535, 562, 634, 706, 778, 895, 913, 922, 958, 985, 1111, 1165, 1219, 1255, 1282, 1507, 1633, 1642, 1678, 1795, 1822, 1858, 1894, 1903, 1921, 1966, 2038, 2155, 2173, 2182, 2218, 2227, 2326, 2362, 2434, 2461, 2515, 2578, 2605, 2614, 2722, 2785, 2839, 2902, 2911, 2965, 2974, 3046, 3091, 3226, 3442, 3505, 3595, 3622, 3649, 3694, 3802, 3865, 3946, 3973, 4054, 4126, 4162, 4189, 4198, 4279, 4306, 4369, 4414, 4594, 4702, 4765, 4855, 4918, 4954, 4981, 5062, 5071, 5098, 5242, 5269, 5305, 5386, 5422, 5458, 5485, 5539, 5602, 5638, 5674, 5818, 5854, 5926, 5935, 5998, 6115, 6178, 6187, 6259, 6295, 6385, 6439, 6457, 6502, 6583, 6718, 6835, 6934, 7051, 7078, 7186, 7195, 7249, 7339, 7402, 7438, 7447, 7465, 7627, 7726, 7762, 7834, 7915, 7978, 8005, 8014, 8023, 8077, 8095, 8149, 8158, 8185, 8257, 8347, 8518, 8545, 8653, 8851, 8914, 9031, 9094, 9166, 9193, 9229, 9274, 9301, 9346, 9355, 9382, 9427, 9535, 9571, 9598, 9634, 9742, 9778, 9895, 9985 etc :

• Palindromic Smith Numbers:

Smith numbers, which are also palindromic ( i.e. reading the same forward as well as backward) can be termed as Palindromic Smith numbers (Sloane's A098834).

All Palindromic Smith numbers below 106 are:

4, 22, 121, 202, 454, 535, 636, 666, 1111, 1881, 3663, 7227, 7447, 9229, 10201, 17271, 22522, 24142, 28182, 33633, 38283, 45054, 45454, 46664, 47074, 50305, 51115, 51315, 54645, 55055, 55955, 72627, 81418, 82628, 83038, 83938, 90409, 95359, 96169, 164461, 173371, 239932, 256652, 262262, 294492, 362263, 373373, 445544, 454454, 505505, 515515, 535535, 545545, 635536, 704407, 717717, 832238, 841148, 864468, 951159, 956659, 974479 and 983389.

• Reversible Smith Numbers:

Smith numbers, whose reversal is also a smith number, can be termed as Reversible Smith numbers (Sloane's A104171).

All Reversible Smith numbers below 104 are:

4, 22, 58, 85, 121, 202, 265, 319, 454, 535, 562, 636, 666, 913, 1111, 1507, 1642, 1881, 1894, 1903, 2461, 2583, 2605, 2614, 2839, 3091, 3663, 3852, 4162, 4198, 4369, 4594, 4765, 4788, 4794, 4954, 4974, 4981, 5062, 5386, 5458, 5539, 5674, 5818, 5926, 6295, 6439, 6835, 7051, 7227, 7249, 7438, 7447, 8158, 8185, 8347, 8518, 8545, 8874, 8914, 9229, 9346, 9355, 9382, 9427 and 9634.

Note that all palindromic smith numbers mentioned above are special case of reversible smith numbers.

• Fibonacci Smith Numbers:

The fibonacci numbers, which are also smith numbers can be termed as Fibonacci Smith Numbers. During computations of smith numbers, it was found that the smallest fibonacci smith number is 1346369.

F31 = 1346269 = 557 x 2417

Since

1+3+4+6+2+6+9 = 5+5+7+2+4+1+7

On further computations two more fibonacci smith numbers have been found, these are:

F77 = 5527939700884757 = 13 x 89 x 988681 x 4832521

F231 = 844617150046923109759866426342507997914076076194
= 2 x 13 x 89 x 421 x 19801 x 988681 x 4832521 x 9164259601748159235188401

It may be interesting study to find more fibonacci smith numbers.

• Abundant and Deficient Smith Numbers:

Numbers such that s(n), the sum of aliquot divisors of n, is greater than n are called Abundant numbers. If n is also a smith number then it can be termed as Abundant Smith Number . For example 438 is a smith number and s(438) = 1 + 2 + 3 + 6 + 73 + 146 + 219 = 450, which is greater than 438. So 438 is a Abundant Smith Number (Sloane's A098835).

All Abundant Smith numbers below 104 are:

378, 438, 576, 588, 636, 648, 654, 666, 690, 728, 762, 852, 1086, 1284, 1376, 1626, 1736, 1776, 1842, 1872, 1908, 1952, 1962, 2286, 2484, 2556, 2576, 2688, 2934, 2944, 2958, 2964, 2970, 3138, 3168, 3174, 3246, 3258, 3294, 3366, 3390, 3564, 3690, 3852, 3864, 3930, 4428, 4464, 4472, 4592, 4788, 4794, 4880, 4960, 4974, 5088, 5172, 5248, 5298, 5388, 5526, 5772, 5874, 5936, 5946, 6036, 6054, 6084, 6096, 6188, 6252, 6344, 6684, 6702, 6760, 6816, 6880, 7026, 7062, 7068, 7674, 7764, 7782, 7784, 7824, 7952, 8154, 8196, 8372, 8412, 8466, 8568, 8628, 8680, 8736, 8754, 8766, 8790, 8792, 8874, 9036, 9184, 9276, 9294, 9296, 9330, 9396, 9414, 9522, 9648, 9684, 9708, 9760, 9840, 9880, 9924, 9942 and 9968.

Numbers such that s(n), the sum of aliquot divisors of n, is less than n are called Deficient numbers. If n is also a smith number then it can be termed as Deficient Smith Number . For example 22 is a smith number and s(22) = 1 + 2 + 11 = 14, which is less than 22. So 22 is a Deficient Smith number (Sloane's A098836).

All Deficient Smith Numbers below 104 are:

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 382, 391, 454, 483, 517, 526, 535, 562, 627, 634, 645, 663, 706, 729, 778, 825, 861, 895, 913, 915, 922, 958, 985, 1111, 1165, 1219, 1255, 1282, 1449, 1507, 1581, 1633, 1642, 1678, 1755, 1795, 1822, 1858, 1881, 1894, 1903, 1921, 1935, 1966, 2038, 2067, 2079, 2155, 2173, 2182, 2218, 2227, 2265, 2326, 2362, 2366, 2373, 2409, 2434, 2461, 2475, 2515, 2578, 2583, 2605, 2614, 2679, 2722, 2745, 2751, 2785, 2839, 2888, 2902, 2911, 2965, 2974, 3046, 3091, 3226, 3345, 3442, 3505, 3595, 3615, 3622, 3649, 3663, 3694, 3802, 3865, 3946, 3973, 4054, 4126, 4162, 4173, 4185, 4189, 4191, 4198, 4209, 4279, 4306, 4369, 4414, 4557, 4594, 4702, 4743, 4765, 4832, 4855, 4918, 4954, 4959, 4981, 5062, 5071, 5098, 5242, 5253, 5269, 5305, 5386, 5397, 5422, 5458, 5485, 5539, 5602, 5638, 5642, 5674, 5818, 5854, 5915, 5926, 5935, 5998, 6115, 6171, 6178, 6187, 6259, 6295, 6315, 6385, 6439, 6457, 6502, 6531, 6567, 6583, 6585, 6603, 6693, 6718, 6835, 6855, 6934, 6981, 7051, 7078, 7089, 7119, 7136, 7186, 7195, 7227, 7249, 7287, 7339, 7402, 7438, 7447, 7465, 7503, 7627, 7683, 7695, 7712, 7726, 7762, 7809, 7834, 7915, 7978, 8005, 8014, 8023, 8073, 8077, 8095, 8149, 8158, 8185, 8253, 8257, 8277, 8307, 8347, 8421, 8518, 8545, 8653, 8851, 8864, 8883, 8901, 8914, 9015, 9031, 9094, 9166, 9193, 9229, 9274, 9285, 9301, 9346, 9355, 9382, 9386, 9387, 9427, 9483, 9535, 9571, 9598, 9633, 9634, 9639, 9657, 9717, 9735, 9742, 9778, 9843, 9849, 9861, 9895, 9975 and 9985.

• Smith Square Numbers:

Smith numbers, which are perfect squares, can be termed as Smith Square Numbers (Sloane's A098839).

All Smith Square Numbers below 107 are:

4, 121, 576, 729, 6084, 10201, 17424, 18496, 36481, 51529, 100489, 124609, 184041, 195364, 410881, 559504, 674041, 695556, 732736, 887364, 896809, 966289, 988036, 1038361, 1190281, 1238769, 1726596, 1852321, 2166784, 2975625, 3407716, 3613801, 3663396, 3849444, 3888784, 3892729, 4088484, 4309776, 4330561, 4809249, 4875264, 4888521, 5031049, 5225796, 5391684, 5438224, 5461569, 5527201, 5978025, 6517809, 6630625, 6635776, 6780816, 6864400, 6969600, 7059649, 7273809, 7717284, 7868025, 7946761, 8048569, 8567329, 8573184, 8608356, 9150625, 9455625, 9678321 and 9960336.

• Smith Cubic Numbers:

Smith numbers, which are perfect cubes, can be termed as Smith Cubic Numbers (Sloane's A098838).

All Smith Cubic Numbers below 1010 are:

27, 729, 19683, 474552, 7077888, 7414875, 8489664, 62099136, 112678587, 236029032, 246491883, 257259456, 279726264, 345948408, 463684824, 567663552, 638277381, 721734273, 766060875, 988047936, 1177583616, 1412467848, 2131746903, 2493326016, 2714704875, 3023464536, 3215578176, 3294646272, 3951805941, 4443297984, 4843965888, 4895680392, 6158676537, 8266914648, 8340725952, 8792838144 and 8831234763.

• Smith Triangular Numbers:

Smith numbers, which are also triangular numbers, can be termed as Smith Triangular Numbers (Sloane's A098840).

All Smith Triangular Numbers below 107 are:

378, 666, 861, 2556, 5253, 7503, 10296, 16653, 27261, 28920, 29890, 32896, 46056, 72771, 84255, 85905, 92235, 94395, 120786, 132870, 141778, 157641, 215496, 328455, 345696, 385881, 386760, 396495, 424581, 529935, 533028, 588070, 654940, 682696, 683865, 723003, 778128, 866586, 885115, 897130, 941878, 959805, 977901, 993345, 1082656, 1134771, 1234806, 1398628, 1457778, 1466328, 1495585, 1528626, 1570878, 1597578, 1633528, 1680861, 1792671, 1875016, 1876953, 1925703, 1931595, 1983036, 2087946, 2089990, 2137278, 2273778, 2351196, 2396955, 2458653, 2692360, 2828631, 2859636, 2890810, 2924571, 2968266, 3296028, 3378700, 3383901, 3451878, 3467661, 3579150, 3654456, 3733278, 3757911, 3904615, 3935415, 4119885, 4444671, 4710915, 4887501, 4925091, 5492955, 5586153, 5592840, 5815755, 6725278, 6984453, 7051890, 7486515, 7505875, 7536903, 7583565, 7850703, 7858630, 8090253, 8158780, 8211378, 8280415, 8284485, 8341570, 8390656, 8398851, 8485140, 8501626, 8982441, 9281586, 9359301, 9541896, 9651421 and 9677800

• Repdigit Smith Numbers:

Smith numbers, whose all digits are same ( i.e. repeated digits) can be termed as Repdigit Smith numbers (Sloane's A104166).

All Repdigit Smith numbers below 1060 are:

4, 22, 666, 1111, 6666666, 4444444444, 44444444444444444444, 555555555555555555555555555, 55555555555555555555555555555555 and 4444444444444444444444444444444444444444444444444444444.

Consecutive Smith Numbers

There are consecutive numbers which are all Smith numbers also, these can be termed as Consecutive Smith numbers. If there are 2 consecutive numbers which are Smith numbers, these are termed as Smith Brothers. The smallest set of 2 consecutive smith numbers i.e. Smith Brothers is (728,729).

All Smith Brothers below 105 are:

(728, 729), (2964, 2965), (3864, 3865), (4959, 4960), (5935, 5936), (6187, 6188), (9386, 9387), (9633, 9634), (11695, 11696), (13764, 13765), (16536, 16537), (16591, 16592), (20784, 20785), (25428, 25429), (28808, 28809), (29623, 29624), (32696, 32697), (33632, 33633), (35805, 35806), (39585, 39586), (43736, 43737), (44733, 44734), (49027, 49028), (55344, 55345), (56336, 56337), (57663, 57664), (58305, 58306), (62634, 62635), (65912, 65913), (65974, 65975), (66650, 66651), (67067, 67068), (67728, 67729), (69279, 69280), (69835, 69836), (73615, 73616), (73616, 73617), (74168, 74169), (74298, 74299), (76495, 76496), (76911, 76912), (77385, 77386), (78744, 78745), (82488, 82489), (82640, 82641), (83744, 83745), (83928, 83929), (83937, 83938), (84759, 84760), (84882, 84883), (85135, 85136), (87362, 87363), (87855, 87856), (89743, 89744), (89904, 89905), (90228, 90229), (90872, 90873), (91255, 91256), (91364, 91365), (91488, 91489), (93275, 93276), (93471, 93472), (94094, 94095), (94184, 94185), (94584, 94585), (95277, 95278), (95984, 95985), (96151, 96152), (96921, 96922), (97915, 97916), (98022, 98023) and (98900, 98901) .

There are 615885 smith brothers below 109.

If there are 3 consecutive numbers which are Smith numbers, these can be termed as Smith Triples. The smallest set of 3 consecutive smith numbers i.e. Smith Triples is (73615,73616,73617).

All Smith triples below 106 are:

(73615, 73616, 73617), (209065, 209066, 209067), (225951, 225952, 225953), (283745, 283746, 283747), (305455, 305456, 305457), (342879, 342880, 342881), (656743, 656744, 656745), (683670, 683671, 683672), (729066, 729067, 729068), (747948, 747949, 747950), (774858, 774859, 774860), (879221, 879222, 879223) and (954590, 954591, 954592).

There are 15955 smith triples below 109.

If there are 4 consecutive numbers which are Smith numbers, these can be termed as Smith Quads (or 4-consecutive Smith numbers). The smallest set of 4-consecutive smith numbers i.e. Smith Quads is (4463535, 4463536, 4463537, 4463538).

Smallest members of set of Smith Quads below 108 are:

4463535, 6356910, 8188933, 9425550, 11148564, 15966114, 18542654, 21673542, 22821992, 23767287, 28605144, 36615667, 39227466, 47096634, 47395362, 48072396, 54054264, 55464835, 57484614, 57756450, 57761165, 58418508, 61843387, 62577157, 64572186, 65484066, 66878432, 67118680, 71845857, 75457380, 77247606, 78432168, 88099213, 89653781, 90166567 and 92656434.

There are 384 smith quads below 109.

If there are 5 consecutive numbers which are Smith numbers, these can be termed as Smith Quints (or 5-consecutive Smith numbers). The smallest set of 5-consecutive smith numbers i.e. Smith Quints is (15966114, 15966115, 15966116, 15966117, 15966118).

Smallest members of set of Smith Quints below 109 are:

15966114, 75457380, 162449165, 296049306, 296861735, 334792990, 429619207, 581097690, 581519244, 582548088, 683474015, 809079150, 971285861 and 977218716 .

There are 14 smith quints below 109.

If there are k consecutive numbers which are Smith numbers, these can be termed as k-consecutive Smith numbers. The smallest set of 6-consecutive smith numbers is (2050918644, 2050918645, 2050918646, 2050918647, 2050918648, 2050918649). There is no set of higher consecutive smith numbers below 1010.

Can you find the smallest set of 7-consecutive smith numbers ?.

Jens Kruse Andersen reported on 28th April 2008 that

The smallest set of 7-consecutive smith numbers starts at 164736913905.

Below that there are 31 sets of 6 consecutive Smith numbers, starting at:

2050918644, 6826932280, 16095667238, 16214788810, 17753840815, 19627891048, 31894287635, 37417358132, 38327645947, 72635842286, 75725224588, 77924458232, 79735902902, 80490527739, 84911527648, 93497450408, 115397414704, 118266684888, 122256909967, 124374538831, 128551622624, 129440489539, 132638590595, 135169942385, 140820590944, 143570578744, 149563926065, 153903366948, 154627494580, 154833907731, 159822348654.

Can you find the smallest set of 8-consecutive smith numbers ?.

k-Smith Numbers

In 1987, Wayne Mc Daniel proved that there are infinitely many Smith numbers[8]. Mc Daniel defined k-Smith numbers as the numbers for which the sum of the digits of the prime factors is equal to k multiplied by sum of the digits i.e. Sp(N)=k*S(N) and showed that there are infinitely many k-Smith numbers for each k by constructing an infinite sequence of them.

For example 42 is a 2-Smith number because digit sum of 42 i.e. S(42) = 4 + 2 = 6, and the sum of the digits of its prime factors i.e. Sp (42) = Sp (2*3*7) = 12. So, Sp(N)=2*S(N).(Sloane's A104390).

All 2-Smith Numbers below 104 are:

32, 42, 60, 70, 104, 152, 231, 315, 316, 322, 330, 342, 361, 406, 430, 450, 540, 602, 610, 612, 632, 703, 722, 812, 1016, 1027, 1029, 1108, 1162, 1190, 1246, 1261, 1304, 1314, 1316, 1351, 1406, 1470, 1510, 1603, 2013, 2054, 2065, 2070, 2071, 2106, 2114, 2121, 2134, 2216, 2230, 2233, 2280, 2322, 2324, 2410, 2413, 2422, 2425, 2506, 2522, 2611, 2701, 2702, 3007, 3030, 3108, 3122, 3130, 3136, 3206, 3212, 3216, 3241, 3302, 3310, 3311, 3412, 3451, 3512, 3560, 3601, 3710, 4025, 4033, 4042, 4080, 4142, 4210, 4440, 4501, 4512, 5002, 5022, 5026, 5073, 5131, 5215, 5222, 5306, 5402, 5900, 6010, 6014, 6031, 6132, 6152, 6201, 6202, 6224, 6251, 6321, 7012, 7016, 7111, 7160, 7202, 7250, 7310, 7410, 7511, 8024, 8232, 8302, 9002, 9030, 9104 and 9322.

There are 2824664 2-Smith Numbers below 109.

If there are composite numbers N such that Sp(N)=3*S(N). These can be termed as 3-Smith numbers (Sloane's A104391).

All 3-Smith Numbers below 105 are:

402, 510, 700, 1113, 1131, 1311, 2006, 2022, 2130, 2211, 2240, 3102, 3111, 3204, 3210, 3220, 4031, 4300, 4410, 5310, 6004, 6100, 6300, 7031, 7120, 9000, 10034, 10125, 10206, 10251, 10304, 10413, 10521, 10612, 10800, 11033, 11111, 11114, 11116, 11121, 11141, 11305, 11520, 11800, 12150, 12240, 12304, 12311, 12320, 12502, 13005, 13022, 13031, 13034, 13223, 13301, 13302, 13320, 13410, 14032, 14141, 14320, 14500, 15022, 15030, 15100, 17101, 20016, 20031, 20052, 20114, 20153, 20213, 20216, 20301, 21030, 21052, 21111, 21250, 21251, 21510, 21511, 21520, 22015, 22202, 22230, 22300, 22410, 23001, 23002, 23022, 23130, 23140, 23212, 23500, 24011, 24021, 24100, 24500, 26011, 30005, 30015, 30051, 30131, 30221, 30303, 30304, 30331, 30413, 31061, 31211, 31230, 31300, 31521, 32004, 32201, 33022, 33100, 33320, 33400, 35100, 36001, 38000, 40040, 40132, 41013, 41400, 41500, 42003, 42100, 42120, 44003, 45030, 50020, 50031, 50032, 51101, 51113, 51120, 51121, 51220, 51400, 52003, 53010, 53400, 60100, 61012, 61201, 62410, 63101, 70021, 71010, 71020, 74000, 80122 and 90100

There are 147982 3-Smith Numbers below 109.

If there are composite numbers N such that Sp(N)=4*S(N). These can be termed as 4-Smith numbers (Sloane's A103125).

All 4-Smith Numbers below 105 are:

2401, 5010, 7000, 10005, 10311, 10410, 10411, 11060, 11102, 11203, 12103, 13002, 13021, 13101, 14001, 14101, 14210, 20022, 20121, 20203, 20401, 21103, 21112, 21120, 21201, 22040, 22101, 22201, 23030, 30003, 30031, 30320, 31002, 31101, 31111, 32101, 32200, 32300, 40002, 41101, 43000, 50001, 50211, 51200, 60001, 61000, 61110, 70002 and 71100 .

There are 13609 4-Smith Numbers below 106.

If there are composite numbers N such that Sp(N)=5*S(N). These can be termed as 5-Smith numbers (Sloane's A103126).

All 5-Smith Numbers below 106 are:

2030, 10203, 12110, 20210, 20310, 21004, 21010, 24000, 24010, 31010, 41001, 50010, 70000, 100004, 100012, 100210, 100310, 100320, 101020, 101041, 102022, 103200, 104010, 104101, 104110, 105020, 106001, 110020, 110202, 110212, 110400, 111013, 112002, 112030, 120010, 120050, 121030, 121201, 122001, 130200, 130210, 132000, 140011, 141010, 200030, 201103, 201111, 201131, 201202, 202003, 202030, 204010, 210210, 211030, 211101, 211120, 211220, 222010, 223010, 300004, 300013, 300300, 300310, 310400, 311001, 311011, 311110, 320010, 321010, 322000, 330020, 350000, 401010, 410002, 410011, 411010, 430000, 500002, 501011, 510110, 600010 and 610000.

There are 4204 5-Smith Numbers below 109.

If there are composite numbers N such that Sp(N)=k*S(N). These can be termed as k-Smith numbers.

For K=6, 7 and 8 , the number of k-Smith numbers. below 109 are 1238, 409 and 218 respectively.

For K=6, 7 and 8 , the smallest k-Smith numbers are 10112, 10 and 200 respectively.

k-1 -Smith Numbers

Mc Daniel [8] defined k-1 -Smith number as composite integer N such that Sp(N)=k-1*S(N) i.e. S(N)=k*Sp(N).

For example 88 is a 2-1 -Smith number because digit sum of 88 i.e. S(88) = 8 + 8=16, which is equal to 2 times the sum of the digits of its prime factors i.e.2 x Sp (88) =2 x Sp (11 x 2 x 2 x 2) = 2 x( 1 + 1 + 2 + 2 + 2) = 16.(Sloane's A050224).

All 2-1 -Smith Numbers below 104 are:

88, 169, 286, 484, 598, 682, 808, 844, 897, 961, 1339, 1573, 1599, 1878, 1986, 2266, 2488, 2626, 2662, 2743, 2938, 3193, 3289, 3751, 3887, 4084, 4444, 4642, 4738, 4804, 4972, 4976, 4983, 5566, 5665, 5764, 5797, 5863, 5876, 6198, 6262, 6541, 6565, 6578, 6655, 6695, 6738, 6868, 6877, 6929, 7278, 7359, 7386, 7458, 7579, 7818, 7843, 7865, 7926, 7953, 7999, 8044, 8086, 8248, 8349, 8446, 8488, 8646, 8824, 8897, 8998, 9269, 9292, 9476, 9696, 9706, 9784, 9889, 9922, 9944 and 9988. .

There are 1087152 2-1 -Smith Numbers below 109.

If there are composite numbers N whose digit sum S(N) is equal to 3 times the sum of the digits of its prime factors i.e. S(N)=3*Sp(N). These can be termed as 3-1 -Smith numbers (Sloane's A050225).

All 3-1 -Smith Numbers below 106 are:

6969, 19998, 36399, 39693, 66099, 69663, 69897, 89769, 99363, 99759, 109989, 118899, 181998, 191799, 199089, 297099, 306939, 333399, 336963, 339933, 363099, 396363, 397998, 399333, 399729, 588969, 606666, 606909, 639633, 660693, 666633, 678666, 693363, 693759, 696333, 759099, 759693, 786666, 787179, 798699, 828759, 834969, 897069, 915969, 928089, 938997, 975963, 990363 and 993729.

There are 6575 3-1 -Smith Numbers below 109.

If there are composite numbers N whose digit sum S(N) is equal to 4 times the sum of the digits of its prime factors i.e. S(N)=4*Sp(N). These can be termed as 4-1 -Smith numbers (Sloane's A103123).

All 4-1 -Smith Numbers below 108 are:

19899699 , 36969999 , 36999699 , 39699969 , 39999399 , 39999993 , 66699699 , 66798798 , 67967799 , 67987986 , 69759897 , 69889389 , 69966699 , 69996993 , 76668999 , 79488798 , 79866798 , 85994799 , 86686886 , 89769759 , 89866568 , 93999993 , 98736968 and 99968199 .

There are 251 4-1 -Smith Numbers below 109.

If there are composite numbers N whose digit sum S(N) is equal to 5 times the sum of the digits of its prime factors i.e. S(N)=5*Sp(N). These can be termed as 5-1 -Smith numbers (Sloane's A103124).

All 5-1 -Smith Numbers below 109 are:

399996663, 666609999, 669969663, 690696969 and 699966663 .

Mc Daniel in his paper[8] conjectured that there exist infinitely many k-1 -Smith numbers for every k >1.

Can you prove the Mc Daniel Conjecture?.

Construction of Smith Numbers

There is no general-purpose formula for generating all Smith numbers.

In 1983, Oltikar and Wayland [12] constructed Smith numbers using repunit primes. For example, 1540 · Rn is a Smith number for each repunit prime Rn. There are a lot of other possible multipliers, including

1540, 1720, 2170, 2440, 5590, 6040, 7930, 8344, 8470, 8920, 23590, 24490, 25228, 29080, 31528, 31780, 33544, 34390, 35380, 39970, 40870, 42490, 42598, 43480, 44380, 45955, 46270, 46810, 46990, 47908, 48790, 49960, 51490, 51625, 52345, 52570, 53290, 57070, 57160, 57880, 59770, 60625, 62146, 63928, 64360, 64540, 65080, 66970, 67528, 69355, 71380, 72982, 73810, 74440, 74710, 76780, 78040, 79030, 79570, 80470, 80740, 82270, 82990, 84790, 85330, 85870, 86590, 87490, 87580, 87850, 88228, 89560, 89740, 89830, 92440, 92620, 95266, 97030, 97048, 98875, 108955 ... (Sloane's A104167).

It has been found that all such multipliers have a digital root of 1 and also have a digital factor sum divisible by 9, for more details click here.

Multiplying any repunit prime Rn >11 by 3304 also produces smith numbers [6][12]. Oltikar and Wayland[12] mentioned that every prime whose digits are all 0 and 1 has some multiple that qualifies as a smith number [2].

In 1984, Pat Costello produced 75 Smith numbers of the form p · q · 10k where p is a small prime and q is a Mersenne prime. The largest Smith number produced was

191 · (2216091 - 1) · 10266 consisting of 65319 digits.

In 1987, Wayne Mc Daniel made a big breakthrough when proving that there are infinitely many Smith numbers[8]. Mc Daniel introduced k-Smith numbers for which the sum of the digits of the prime factors is equal to k multiplied by sum of the digits and showed that there are infinitely many k-Smith numbers for each k by constructing an infinite sequence of them.

Kathy Lewis [2] produced another infinite sequence of Smith numbers of the form 11k · 9 Rn · 10m.

Distribution of Smith Numbers v/s Primes

A computer program in Fortran has been developed to investigate Smith numbers. Objective of investigation was to know the proportion of Smith numbers as compared to prime numbers up to a given limit and maximum number of prime factors of Smith numbers. Let the number of Smith numbers up to x is denoted by SN(x) and number of primes up to x by pi(x). The values of SN(x)and pi(x) up to x = 10m for m = 1,2,3, . . 11 are tabulated in Table 1. The ratio r = pi(x)/SN(x) is tabulated in column 4 of Table 1.

It is noted from Table 1 that percentage of Smith numbers decreases with increase in x. As an example, for x = 105 , percentage of Smith numbers up to x = 3.294% which decreases to 2.9928% for x = 106 . The percentage of primes also decreases with increase in x. It is observed that the ratio r decreases with increase in x for x = 10m(for m>2), which indicate that the decrease in Smith numbers is at a slower rate, then decrease in prime numbers.

Table 1

 x SN(x) pi(x) pi(x)/ SN(x) 101 1 4 4.00000 102 6 25 4.16667 103 49 168 3.42857 104 376 1229 3.26862 105 3294 9592 2.91196 106 29928 78498 2.62289 107 278411 664579 2.38704 108 2632758 5761455 2.18837 109 25154060 50847534 2.02144 1010 241882509 455052511 1.88129 1011 2335807857 4118054813 1.76301

It is also noted during investigation that distribution of Smith numbers between 10m and 10m+1 generally presents a rising trend in this range. So generally SN(2 x 10m ) - SN(10m ) is lower than SN (10m+1 ) - SN(9 x 10m ).

As an example: for m = 7,

SN(2 x 107 ) - SN (107 ) = 527739 - 278411 = 249328

SN(108 ) - SN(9 x 107 ) = 2632758 - 2353482 = 279276

for m = 8,

SN(2 x 108 ) - SN(108 ) = 5029891 - 2632758 = 2397133

SN(109 ) - SN (9 x 108 ) = 25154060 - 22497704 = 2656356

This behavior of distribution of Smith numbers is different from prime numbers, which shows a falling trend. So it is likely that for large x, SN(x) may exceed pi(x) or in other words the ratio r may be less than 1. So we state the following Conjecture.

Conjecture: There exists a number x up to which, number of Smiths are equal to the number of Primes and also SN(10m) > pi(10m) for a value of m such that 10m > x .

It may be an interesting study to find smallest value of x (which may be quite large, more than 1018) for which SN(x) = pi(x).

Highly Decomposable Smith Numbers

Let the number of prime factors (counting multiplicity) of a number n is denoted by Ω(n), then n can be termed as Highly Decomposable number if for every n1 < n, Ω(n1) < Ω(n). It is obvious that the ith Highly Decomposable number is 2i where, i=1,2,3,4...

Similarly , for every Smith number N1 < N, if Ω(N1) < Ω(N), then N can be termed as Highly Decomposable Smith number [4]. In other words, the Smith number which sets a record for number of prime factors, starting from the first Smith number can be termed as Highly Decomposable Smith number (Sloane's A104169). The Smallest Smith numbers with n prime factors (not necessarily distinct) are tabulated in Table 2(Sloane's A104168). It is found that the maximum number of prime factors of a Smith number up to 1010 is 27.

As can be seen from Table 2, the sequence of Highly Decomposable Smith numbers is

4, 27, 378, 576, 2688, 17496, 44928, 75776, 168960, 319488, 958464, 2883584, 5767168, 7077888, 279969792, 544997376, 778567680, 2579496960, 3875536896 . . .

It may be noted that except the number 27, all other Highly Decomposable Smith numbers are even.

Table 2

 No. of Prime factors Smallest Smith number 2 4 3 27 4 636 5 378 6 729 7 648 8 576 9 2688 10 17496 11 44928 12 75776 13 168960 14 765952 15 319488 16 958464 17 5537792 18 5963776 19 2883584 20 5767168 21 7077888 22 279969792 23 544997376 24 778567680 25 2579496960 26 4567597056 27 3875536896

Conjecture: All Highly Decomposable Smith numbers greater than 27 are even.

It may be interesting to find a counter example or to prove the above conjecture.

Some Interesting Observations

• 31 is the smallest prime which is the sum of two smith numbers.
• Multiplying a number by 10 does not change the digit sum but add 7( = 2+5) to the sum of the digits of its prime factors.
• Smallest and largest Pandigital (i.e. containing all digits from 0 to 9) smith numbers are 1023465798 and 9876542103.
• Smallest and largest zeroless Pandigital (i.e. containing all digits from 1 to 9) smith numbers are 123456879 and 987653214.
• 728729 is a prime formed from concatenation of two consecutive smith numbers.
• 9R1031*(1028572+8*1014286+1)1027 * 102722434 is one of the largest known smith numbers consisting of 32066910 digits found by Patrick Costello[1].
• 9R1031 *(1069882+3*1034941+1)1476*103913210 is a smith number consisting of 107060074 digits.
• If m and n are positive integers , then Sp(mn) = Sp(m) + Sp(n) where Sp(n) denote the sum of digits of the prime factors of n. So, Sp is an additive function.
• A smith number with a digital root other than 4 must be the product of more than two primes.
• 2227 is the smallest smith number formed from concatenation of two consecutive smith numbers.
• 27, 58 and 85 are three consecutive Smith numbers and 27 + 58 = 85 .

-----------------------------------------------------------------------------------------------------

[1] Costello, Patrick. "A New Largest Smith Number," Fibonacci Quarterly, vol 40(4), 2002, pp. 369-371.

[2] Costello, Patrick and Lewis, Kathy. "Lots of Smiths," Mathematics Magazine, vol 75(3), 2002, pp. 223-226.

[3] Dudley,U. , "Smith Numbers", Mathematics Magazine, 67(1994),pp.62-65.

[4] Gupta, Shyam Sunder "Smith Numbers" Mathematical Spectrum ,37(2004/5), pp.27-29.

[5] Guy, R. K. "Smith Numbers." §B49 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 103-104, 1994.

[6] Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 205-206, 1998.

[7] McDaniel,W.L., "Palindromic Smith numbers", Journal of Recreational Mathematics, 19(1987), pp. 34-37.

[8] McDaniel,W.L., "The Existence of infinitely Many k- Smith numbers", Fibonacci Quarterly, 25(1987), pp.76-80.

[9] McDaniel,W.L., "Powerful k-Smith Numbers", Fibonacci Quarterly, 25(1987), pp.225-228.

[10] McDaniel,W.L. and Yates, Samuel, "The Sum of Digits Function and its Application to A Generalization of the Smith Number Problem", Nieuw Archief Voor Wiskunde, 7, No, 1-2, March/July, (1989), 39-51.

[11] McDaniel,W.L., "On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers", Missouri J. of Math. Sci. 2 (1990), 132-136.

[12] Oltikar, Sham and Keith Wayland. "Construction of Smith Number," Mathematics Magazine, vol 56(1), 1983,
pp. 36-37.

[13] Pickover, Clifford A. "A Brief History of Smith Numbers." Ch. 104 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford University Press, pp. 247-248, 2001.

[14] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

[15] Sloane, N. J. A. Sequences A006753, A033662, A033663, A050218, A050224, A050225, A059754, A063844, A098834, A098835, A098836, A098837, A098838, A098839, A098840, A103123, A103124, A103125, A103126, A104166, A104167, A104168, A104169, A104170, A104171, A104390 and A104391, in "The On-Line Encyclopedia of Integer Sequences."

[16] Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1997.

[17] Wilansky, A. "Smith numbers",Two-Year College Mathematics Journal, 13(1982),p.21.

[18] Yates,S., "Smith numbers congruent to 4 (mod 9)", Journal of Recreational Mathematics, 19(1987), pp.139-141.