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A344512 a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.

%I A344512 #6 Feb 16 2025 08:34:02
%S A344512 4,13,13,13,287,287,2971,2971,27163,27163,90163,90163,5940609,5940609,
%T A344512 6069129,6069129,276404649,276404649
%N A344512 a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.
%C A344512 Since the sequence of base-b self numbers for odd b is the sequence of the odd numbers (A005408) (Joshi, 1973), all the terms beyond a(2) are odd numbers.
%C A344512 For the corresponding sequence with only even bases, see A344513.
%C A344512 a(20) > 1.5*10^10, if it exists.
%D A344512 Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
%D A344512 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
%H A344512 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SelfNumber.html">Self Number</a>.
%H A344512 Wikipedia, <a href="http://en.wikipedia.org/wiki/Self_number">Self number</a>.
%H A344512 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A344512 a(2*n+1) = a(2*n) for n >= 2.
%e A344512 a(2) = 4 since the least binary self number after 1 is A010061(2) = 4.
%e A344512 a(3) = 13 since the least binary self number after 1 which is also a self number in base 3 is A010061(4) = 13.
%t A344512 s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[2] = 4; a[b_] := a[b] = Module[{n = a[b - 1]}, While[! AllTrue[Range[2, b], selfQ[n, #] &], n++]; n]; Array[a, 10, 2]
%Y A344512 Cf. A003052, A010061, A010064, A010067, A010070, A339211, A339212, A339213, A339214, A339215, A342729, A344513.
%Y A344512 Similar sequences: A016038, A217705, A225427, A226320, A228768, A258107.
%K A344512 nonn,base,more
%O A344512 2,1
%A A344512 _Amiram Eldar_, May 21 2021