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 A178193 #28 Feb 16 2025 08:33:12
%S A178193 3777,7773,17418,30777,53921,66111,97731,111916,119217,122519,128131,
%T A178193 133195,135488,138878,145229,178814,180174,198581,257376,269636,
%U A178193 281179,296396,317686,358256,362996,366514,394114,435777,457377,469552,475856,502960,513833
%N A178193 Smith numbers of order 4.
%C A178193 Composite numbers n not in A176670 such that the sum of the 4th power of the digits of n equals the sum of the 4th power 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 A178193 Donovan Johnson, <a href="/A178193/b178193.txt">Table of n, a(n) for n = 1..1000</a>
%H A178193 Patrick Costello, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-4/costello.pdf">A new largest Smith number</a>, Fibonacci Quarterly 40(4) (2002), 369-371.
%H A178193 Underwood Dudley, <a href="https://www.jstor.org/stable/2690561">Smith numbers</a>, Mathematics Magazine 67(1) (1994), 62-65.
%H A178193 S. S. Gupta, <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol37_No1.pdf">Smith Numbers</a>, Mathematical Spectrum 37(1) (2004/5), 27-29.
%H A178193 S. S. Gupta, <a href="http://www.shyamsundergupta.com/smith.htm">Smith Numbers</a>.
%H A178193 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmithNumber.html">Smith number</a>.
%H A178193 Wikipedia, <a href="https://en.wikipedia.org/wiki/Smith_number">Smith number</a>.
%H A178193 A. Wilansky, <a href="https://www.jstor.org/stable/3026531">Smith Numbers</a>, Two-Year College Math. J. 13(1) (1982), p. 21.
%H A178193 Amin Witno, <a href="https://projecteuclid.org/euclid.mjms/1312233139">Another simple construction of Smith numbers</a>, Missouri J. Math. Sci. 22(2) (2010), 97-101.
%H A178193 Amin Witno, <a href="http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/952">Smith multiples of a class of primes with small digital sum</a>, Thai Journal of Mathematics 14(2) (2016), 491-495.
%e A178193 3777 = 3*1259 is composite; sum of 4th power of the digits is 3^4 + 7^4 + 7^4 + 7^4 = 7284. Sum of 4th power of the digits of the prime factors 3, 1259 is 3^4 + 1^4 + 2^4 + 5^4 + 9^4 = 7284. The sums are equal, so 3777 is in the sequence.
%e A178193 17418 = 2*3*2903 is composite; sum of 4th power of the digits is 1^4 + 7^4 + 4^4 + 1^4 + 8^4 = 6755. Sum of 4th power of the digits of the prime factors 2, 3, 2903 is 2^4 + 3^4 + 2^4 + 9^4 + 0^4 + 3^4 = 6755. The sums are equal, so 17418 is in the sequence.
%e A178193 269636 = 2*2*67409 is composite; sum of 4th power of the digits is 2^4 + 6^4 + 9^4 + 6^4 + 3^4 + 6^4 = 10546. Sum of 4th power of the digits of the prime factors 2, 2, 67409 (with multiplicity) is 2^4 + 2^4 + 6^4 + 7^4 + 4^4 + 0^4 + 9^4 = 10546. The sums are equal, so 269636 is in the sequence.
%t A178193 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]]; id != fid && Plus @@ (id^4) == Plus @@ (fid^4)]; k = 1; lst = {}; While[k < 10^6, If[f Q@ k, AppendTo[lst, k]; Print@ k]; k++]; lst
%Y A178193 Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178203, A178204.
%K A178193 nonn,base
%O A178193 1,1
%A A178193 _Paul Weisenhorn_, Dec 19 2010