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A324858 Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.

%I A324858 #28 Oct 06 2024 18:14:09
%S A324858 28,40,52,66,76,88,96,100,112,120,126,136,148,153,156,160,176,186,190,
%T A324858 196,208,225,232,246,268,276,280,288,292,297,304,306,328,336,340,344,
%U A324858 352,366,369,370,378,388,396,400,408,435,441,448,456,460,486,496,513,516,520,532,540,544,546,550,560,568,576,580,585,592
%N A324858 Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.
%C A324858 The function s_c(m) gives the sum of the base-c digits of m.
%C A324858 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 subsequence A324857 = numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p. However, d is usually composite for higher values of m.
%C A324858 For any composite c, 0 < b < c, and 0 < i < j, b*c^i + (c-b)*c^j is in the sequence. - _Robert Israel_, Mar 19 2019
%C A324858 The sequence does not contain the 3-Carmichael numbers A087788, but intersects the Carmichael numbers A002997 that have at least four factors. This is a nontrivial fact. Examples for such Carmichael numbers below one million: 41041 = 7*11*13*41, 172081 = 7*13*31*61, 188461 = 7*13*19*109, 278545 = 5*17*29*113, 340561 = 13*17*23*67, 825265 = 5*7*17*19*73. For further properties of the terms see A324456 and Kellner 2019. - _Bernd C. Kellner_, Apr 02 2019
%H A324858 Robert Israel, <a href="/A324858/b324858.txt">Table of n, a(n) for n = 1..10000</a>
%H A324858 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 A324858 s_4(28) = 4 as 28 = 3 * 4 + 1 * 4^2, so 28 is a member.
%p A324858 S:= proc(c,m) convert(convert(m,base,c),`+`) end proc:
%p A324858 filter:= proc(m) ormap(c -> (S(c,m)=c), remove(isprime,numtheory:-divisors(m) minus {1})) end proc:
%p A324858 select(filter, [$1..1000]); # _Robert Israel_, Mar 19 2019
%t A324858 s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
%t A324858 f[n_] := AnyTrue[Divisors[n], CompositeQ[#] && s[n, #] == # &];
%t A324858 Select[Range[600], f[#] &] (* simplified by _Bernd C. Kellner_, Apr 02 2019 *)
%o A324858 (PARI) isok(n) = {fordiv(n, d, if ((d>1) && !isprime(d) && (sumdigits(n, d) == d), return (1)););} \\ _Michel Marcus_, Mar 19 2019
%Y A324858 A324456 is the union of A324857 and A324858.
%K A324858 nonn,base
%O A324858 1,1
%A A324858 _Jonathan Sondow_, Mar 17 2019