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A324455 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) >= g.

%I A324455 #25 Oct 05 2024 16:29:42
%S A324455 6,10,12,14,15,18,20,21,22,24,26,28,30,33,34,36,38,39,40,42,44,45,46,
%T A324455 48,50,51,52,54,56,57,58,60,62,63,65,66,68,69,70,72,74,75,76,78,80,82,
%U A324455 84,85,86,87,88,90,91,92,93,94,95,96,98,99,100,102,104,105
%N A324455 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) >= g.
%C A324455 The function s_g(m) gives the sum of the base-g digits of m.
%C A324455 The sequence is infinite, since it contains A324460 and the Carmichael numbers A002997.
%C A324455 A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
%C A324455 See Kellner 2019.
%H A324455 Bernd C. Kellner, <a href="/A324455/b324455.txt">Table of n, a(n) for n = 1..743</a>
%H A324455 Bernd C. Kellner, <a href="https://doi.org/10.5281/zenodo.10963985">On primary Carmichael numbers</a>, Integers 22 (2022), Article #A38, 39 pp.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.
%e A324455 6 is a member, since 2 divides 6 and s_2(6) = 2.
%t A324455 s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
%t A324455 f[n_] := AnyTrue[Divisors[n], s[n, #] >= # &];
%t A324455 Select[Range[1000], f[#] &]
%Y A324455 Subsequences are A002997, A324315, A324316, A324456, A324457, A324458, A324459, A324460.
%K A324455 nonn,base
%O A324455 1,1
%A A324455 _Bernd C. Kellner_, Feb 28 2019