Smith numbers have the property that the sums of their digits are equal to the sums of the digits of their prime factors with multiplicity. Based on that criterion, the number associated with my diurnal age today, 28174, is a Smith number:$$28174=2 \times 14087$$This number however, has a further property if you look at it closely. The number and its prime factors share the same digits with the exception of the zero. This qualifies it for membership in OEIS A176670:
A176670: composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.
The initial members of the sequence are:
1111, 1255, 12955, 17482, 25105, 28174, 51295, 81229, 91365, 100255, 101299, 105295, 107329, 110191, 110317, 117067, 124483, 127417, 129595, 132565, 137281, 145273, 146137, 149782, 163797, 171735, 174082, 174298, 174793, 174982, 193117, 208174, 210181, 217894
The table below shows the details:
number digit sum factors sum of factors' digits 1111 4 11 * 101 4 1255 13 5 * 251 13 12955 22 5 * 2591 22 17482 22 2 * 8741 22 25105 13 5 * 5021 13 28174 22 2 * 14087 22 51295 22 5 * 10259 22 81229 22 29 * 2801 22 91365 24 3 * 5 * 6091 24 100255 13 5 * 20051 13 101299 22 11 * 9209 22 105295 22 5 * 21059 22 107329 22 29 * 3701 22 110191 13 101 * 1091 13 110317 13 107 * 1031 13 117067 22 167 * 701 22 124483 22 281 * 443 22 127417 22 47 * 2711 22 129595 31 5 * 25919 31 132565 22 5 * 26513 22 137281 22 107 * 1283 22 145273 22 53 * 2741 22 146137 22 317 * 461 22 149782 31 2 * 74891 31 163797 33 3 * 71 * 769 33 171735 24 3 * 5 * 107^2 24 174082 22 2 * 87041 22 174298 31 2 * 87149 31 174793 31 47 * 3719 31 174982 31 2 * 87491 31 193117 22 113 * 1709 22 208174 22 2 * 104087 22 210181 13 101 * 2081 13 217894 31 2 * 108947 31What's interesting is that these same digits show up several times in the above table:
- 17482 which is a permutation of the digits of 28174
- 149782 which has the digit 9 added to the permuted digits
- 174082 which has the digit 0 added to the permuted digits
- 174298 which has the digit 9 added to the permuted digits
- 174982 which has the digit 9 added to the permuted digits
- 208174 which has the digit 0 inserted after the 2 in 28174
All these numbers are xenodromes meaning that they have no repeated digits.
The obverse of this is to find all Smith numbers that have NO digits in common with their prime factors. There are 72 of these in the range up 40000 (permalink):
4, 27, 58, 166, 454, 576, 588, 627, 648, 654, 666, 690, 706, 729, 1449, 1858, 1908, 2067, 2409, 2839, 4369, 4414, 4464, 4880, 4960, 5458, 5818, 5854, 6084, 6096, 6567, 6583, 6684, 6718, 6760, 6880, 7068, 7078, 7186, 8158, 8568, 8680, 8864, 8901, 9166, 9414, 9849, 10669, 10786, 10966, 14458, 14566, 14958, 15646, 15709, 15984, 16546, 16866, 17496, 17664, 17718, 17840, 18418, 18454, 19818, 20229, 20299, 22509, 26727, 33680, 33760, 33880
The table below shows the details:
number digit sum factors sum of factors' digits 4 4 2^2 4 27 9 3^3 9 58 13 2 * 29 13 166 13 2 * 83 13 454 13 2 * 227 13 576 18 2^6 * 3^2 18 588 21 2^2 * 3 * 7^2 21 627 15 3 * 11 * 19 15 648 18 2^3 * 3^4 18 654 15 2 * 3 * 109 15 666 18 2 * 3^2 * 37 18 690 15 2 * 3 * 5 * 23 15 706 13 2 * 353 13 729 18 3^6 18 1449 18 3^2 * 7 * 23 18 1858 22 2 * 929 22 1908 18 2^2 * 3^2 * 53 18 2067 15 3 * 13 * 53 15 2409 15 3 * 11 * 73 15 2839 22 17 * 167 22 4369 22 17 * 257 22 4414 13 2 * 2207 13 4464 18 2^4 * 3^2 * 31 18 4880 20 2^4 * 5 * 61 20 4960 19 2^5 * 5 * 31 19 5458 22 2 * 2729 22 5818 22 2 * 2909 22 5854 22 2 * 2927 22 6084 18 2^2 * 3^2 * 13^2 18 6096 21 2^4 * 3 * 127 21 6567 24 3 * 11 * 199 24 6583 22 29 * 227 22 6684 24 2^2 * 3 * 557 24 6718 22 2 * 3359 22 6760 19 2^3 * 5 * 13^2 19 6880 22 2^5 * 5 * 43 22 7068 21 2^2 * 3 * 19 * 31 21 7078 22 2 * 3539 22 7186 22 2 * 3593 22 8158 22 2 * 4079 22 8568 27 2^3 * 3^2 * 7 * 17 27 8680 22 2^3 * 5 * 7 * 31 22 8864 26 2^5 * 277 26 8901 18 3^2 * 23 * 43 18 9166 22 2 * 4583 22 9414 18 2 * 3^2 * 523 18 9849 30 3 * 7^2 * 67 30 10669 22 47 * 227 22 10786 22 2 * 5393 22 10966 22 2 * 5483 22 14458 22 2 * 7229 22 14566 22 2 * 7283 22 14958 27 2 * 3^3 * 277 27 15646 22 2 * 7823 22 15709 22 23 * 683 22 15984 27 2^4 * 3^3 * 37 27 16546 22 2 * 8273 22 16866 27 2 * 3^2 * 937 27 17496 27 2^3 * 3^7 27 17664 24 2^8 * 3 * 23 24 17718 24 2 * 3 * 2953 24 17840 20 2^4 * 5 * 223 20 18418 22 2 * 9209 22 18454 22 2 * 9227 22 19818 27 2 * 3^3 * 367 27 20229 15 3 * 11 * 613 15 20299 22 53 * 383 22 22509 18 3^2 * 41 * 61 18 26727 24 3 * 59 * 151 24 33680 20 2^4 * 5 * 421 20 33760 19 2^5 * 5 * 211 19 33880 22 2^3 * 5 * 7 * 11^2 22
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