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%I #35 Dec 05 2020 04:08:02
%S 6,10,12,15,18,20,21,24,28,33,34,36,39,40,45,48,52,57,63,65,66,68,72,
%T 76,80,85,87,88,91,93,96,99,100,105,111,112,117,120,126,130,132,133,
%U 135,136,144,145,148,153,156,160,165,171,175,176,185,186,189,190
%N Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.
%C The function s_g(m) gives the sum of the base-g digits of m.
%C The sequence is infinite, since it contains A324460.
%C The sequence also contains the 3-Carmichael numbers A087788 and the primary Carmichael numbers A324316.
%C 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 Note that the sequence contains the 3-Carmichael numbers, but not all Carmichael numbers. This is a nontrivial fact.
%C The subsequence A324460 mainly gives examples in which g is composite.
%C See Kellner 2019.
%C It appears that g is usually prime: compare with A324857 (g prime) and the sparser sequence A324858 (g composite). However, g is usually composite for higher values of m. - _Jonathan Sondow_, Mar 17 2019
%H Amiram Eldar, <a href="/A324456/b324456.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..806 from Bernd C. Kellner)
%H Bernd C. Kellner, <a href="https://arxiv.org/abs/1902.11283">On primary Carmichael numbers</a>, arXiv:1902.11283 [math.NT], 2019.
%H Bernd C. Kellner and Jonathan Sondow, <a href="https://arxiv.org/abs/1902.10672">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, arXiv:1902.10672 [math.NT], 2019.
%e 6 is a member, since 2 divides 6 and s_2(6) = 2.
%t s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
%t f[n_] := AnyTrue[Divisors[n], s[n, #] == # &];
%t Select[Range[5000], f[#] &]
%o (PARI) isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)););} \\ _Michel Marcus_, Mar 19 2019
%Y Subsequences are A033502, A087788, A324316, A324458, A324460.
%Y Subsequence of A324455.
%Y Cf. A324315, A324457, A324459.
%Y Union of A324857 and A324858.
%K nonn,base
%O 1,1
%A _Bernd C. Kellner_, Feb 28 2019
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