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A324857 Numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p.

%I A324857 #28 Oct 06 2024 18:14:13
%S A324857 6,10,12,15,18,20,21,24,33,34,36,39,40,45,48,57,63,65,66,68,72,80,85,
%T A324857 87,91,93,96,99,105,111,117,130,132,133,135,136,144,145,160,165,171,
%U A324857 175,185,189,192,205,217,225,231,249,255,258,259,260,261,264,265,272,273,279,285,288,297,301,305,320,325,327,333,341,351,384,385
%N A324857 Numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p.
%C A324857 The function s_p(m) gives the sum of the base-p digits of m.
%C A324857 m must have at least 2 prime factors, since s_p(p^k) = 1 < p.
%C A324857 The sequence contains the primary Carmichael numbers A324316.
%C A324857 The main entry for this sequence is A324456 = numbers m > 1 such that there exists a divisor d > 1 of m with s_d(m) = d. It appears that d is usually prime: compare the sparser sequence A324858 = numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c. However, d is usually composite for higher values of m.
%C A324857 The sequence contains the 3-Carmichael numbers A087788, but not all Carmichael numbers A002997. This is a nontrivial fact. The smallest Carmichael number that is not a member is 173085121 = 11*31*53*61*157. For further properties of the terms see A324456 and Kellner 2019. - _Bernd C. Kellner_, Apr 02 2019
%H A324857 Robert Israel, <a href="/A324857/b324857.txt">Table of n, a(n) for n = 1..10000</a>
%H A324857 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 A324857 s_p(m) = 1 < p for m = 2, 3, 4, 5 with prime p dividing m, but if m = 6 and p = 2 then s_p(m) = s_2(2 + 2^2) = 1 + 1 = 2 = p, so a(1) = 6.
%p A324857 S:= (p,m) -> convert(convert(m,base,p),`+`):
%p A324857 filter:= proc(m) ormap(p -> S(p,m) = p, numtheory:-factorset(m)) end proc:
%p A324857 select(filter, [$2..500]); # _Robert Israel_, Mar 20 2019
%t A324857 s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
%t A324857 f[n_] := AnyTrue[Divisors[n], PrimeQ[#] && s[n, #] == # &];
%t A324857 Select[Range[400], f[#] &]n (* simplified by _Bernd C. Kellner_, Apr 02 2019 *)
%o A324857 (PARI) isok(n) = {if (n>1, my(vp=factor(n)[,1]); for (k=1, #vp, if (sumdigits(n, vp[k]) == vp[k], return (1)))); } \\ _Michel Marcus_, Mar 19 2019
%Y A324857 A324456 is the union of A324857 and A324858.
%Y A324857 Includes A083558.
%K A324857 nonn,base
%O A324857 1,1
%A A324857 _Jonathan Sondow_, Mar 17 2019