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 A357131 #58 Feb 16 2025 08:34:04
%S A357131 0,137,11126,111134,111278,1111223,11111447,111112247,1111122227,
%T A357131 111111111137,11111111111126,111111111111134,1111111111111223,
%U A357131 111111111111111111111111111111111111111111111111111111111111111111111111111111111111278
%N A357131 Numbers m such that A010888(m) = A031347(m) = A031286(m) = A031346(m); only the least of the anagrams are considered.
%C A357131 The infinite sequence {(R_1)37, (R_91)37, (R_991)37, (R_9991)37, (R_99991)37, ...} is a subsequence, where R_k is the repunit of length k. Hence this sequence is infinite.
%C A357131 a(14) <= (R_84)278.
%C A357131 Some additional terms < 10^100: (R_84)278, (R_86)447, (R_86)2247, (R_86)22227, (R_91)37, (R_93)26, (R_94)34, (R_93)278, (R_94)223, (R_95)447, (R_95)2247, (R_95)22227.
%C A357131 If a term k > 0 then k cannot contain a digit 0 as if it does A031347(k) = 0 while A031346(k) = 1, contradicting equality. - _David A. Corneth_, Sep 15 2022
%C A357131 a(14) > 10^50. - _Michael S. Branicky_, Sep 16 2022
%C A357131 From _Michael S. Branicky_, Sep 17 2022: (Start)
%C A357131 (R_{10^k})37 and (R_{2*10^k - 10})37 also form infinite subsequences for k >= 0.
%C A357131 Indeed, terms of the form (R_k)e form infinite subsequences for each e in 26, 34, 37, 223, 278, 447, 2247, 22227 for k such that A007953(k + A007953(e)) = 2.
%C A357131 a(2)-a(14) and all terms of the forms above have 2 = A010888(m) = A031347(m) = A031286(m) = A031346(m).
%C A357131 (R_{t})277 where t+2+7+7 = 4 followed by 55555 9's is a term with 4 = A010888(m) = A031347(m) = A031286(m) = A031346(m).
%C A357131 Likewise, there exists a term of the form (R_{t})5579 with 5 = A010888(m) = A031347(m) = A031286(m) = A031346(m), where t+26 is part of the additive persistence chain ending ..., 5999999, 59, 14, 5. Likewise for 888899 and 6, and so on.
%C A357131 However, there are no terms with A010888(m) = A031347(m) = A031286(m) = A031346(m) = 1 or 3. (End)
%H A357131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AdditivePersistence.html">Additive Persistence</a>
%H A357131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>
%H A357131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativeDigitalRoot.html">Multiplicative Digital Root</a>
%H A357131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%e A357131 137 is in the sequence as A010888(137) = 137 mod 9 = 2, A031347(137) = 2 via 1*3*7 = 21 -> 2*1 = 2 < 10, A031286(137) = 2 via 1+3+7 = 11 -> 1+1 = 2 so two steps, A031346(137) = 2 via 1*3*7 = 21 -> 2*1 = 2 so two steps. As all these outcomes are 2, 137 is a term. - _David A. Corneth_, Sep 15 2022
%Y A357131 Subsequence of A179239.
%Y A357131 Cf. A010888, A031347, A031286, A031346.
%Y A357131 Cf. A064702, A239427.
%K A357131 nonn,base
%O A357131 1,2
%A A357131 _Mohammed Yaseen_, Sep 14 2022
%E A357131 a(8)-a(13) from _Pontus von Brömssen_, Sep 14 2022
%E A357131 a(14) confirmed by _Michael S. Branicky_, Sep 17 2022