A324456 - OEIS

%I #39 Oct 05 2024 16:29:54

%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://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.

%H Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.5281/zenodo.10816833">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), Article #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [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