This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A178213 #40 Feb 16 2025 08:33:12
%S A178213 6606,8540,13086,16866,21080,26637,27468,33387,34790,35364,35377,
%T A178213 40908,44652,48154,48860,52798,54814,55055,57726,57894,66438,67297,
%U A178213 67356,67594,69549,72465,72598,73026,74371,74785,77485,78745,81546,83175,85927,90174,91208
%N A178213 Smith numbers of order 3.
%C A178213 Composite numbers n not in A176670 such that the sum of the cubes of the digits of n equals the sum of the cubes of the digits of the prime factors of n (with multiplicity). A176670 lists composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.
%H A178213 Ely Golden and Donovan Johnson, <a href="/A178213/b178213.txt">Table of n, a(n) for n = 1..10000</a> (terms 1 to 1000 by Donovan Johnson)
%H A178213 Ely Golden, <a href="/A178213/a178213_1.sagews.txt">Smith number sequence generator optimized for order 3</a>
%H A178213 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/SmithNumber.html">Smith Number</a>
%e A178213 6606 = 2*3*3*367 is composite; sum of cubes of the digits is 6^3+6^3+0^3+6^3 = 648. Sum of cubes of the digits of the prime factors 2, 3, 3, 367 (with multiplicity) is 2^3+3^3+3^3+3^3+6^3+7^3 = 648. The sums are equal, so 6606 is in the sequence.
%e A178213 21080 = 2*2*2*5*17*31 is composite; sum of cubes of the digits is 2^3+1^3+0^3+8^3+0^3 = 521. Sum of cubes of the digits of the prime factors 2, 2, 2, 5, 17, 31 (with multiplicity) is 2^3+2^3+2^3+5^3+1^3+7^3+3^3+1^3 = 521. The sums are equal, so 21080 is in the sequence.
%t A178213 fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; !PrimeQ@ n && id != fid && Plus @@ (id^3) == Plus @@ (fid^3)]; k = 2; lst = {}; While[k < 22002, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst
%t A178213 With[{k = 3}, Select[Range[10^5], Function[n, And[Total@ Map[#^k &, IntegerDigits@ n] == Total@ Map[#^k &, Flatten@ IntegerDigits[#]], Not[Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@#]] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* _Michael De Vlieger_, Dec 10 2016 *)
%Y A178213 Cf. A006753 (Smith numbers), A174460, A176670, A178193, A178203, A178204.
%K A178213 nonn,base
%O A178213 1,1
%A A178213 _Paul Weisenhorn_, Dec 19 2010