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

28, 40, 52, 66, 76, 88, 96, 100, 112, 120, 126, 136, 148, 153, 156, 160, 176, 186, 190, 196, 208, 225, 232, 246, 268, 276, 280, 288, 292, 297, 304, 306, 328, 336, 340, 344, 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

Offset: 1

Author

Jonathan Sondow, Mar 17 2019

Keywords

• nonn
• base

Comments

The function s_c(m) gives the sum of the base-c digits of m.

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.

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

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

Examples

			s_4(28) = 4 as 28 = 3 * 4 + 1 * 4^2, so 28 is a member.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Crossrefs

A324456 is the union of A324857 and A324858.

Programs

Maple

S:= proc(c,m) convert(convert(m,base,c),`+`) end proc:
filter:= proc(m) ormap(c -> (S(c,m)=c), remove(isprime,numtheory:-divisors(m) minus {1})) end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

Mathematica

s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
f[n_] := AnyTrue[Divisors[n], CompositeQ[#] && s[n, #] == # &];
Select[Range[600], f[#] &] (* simplified by Bernd C. Kellner, Apr 02 2019 *)

PARI

isok(n) = {fordiv(n, d, if ((d>1) && !isprime(d) && (sumdigits(n, d) == d), return (1)););} \\ Michel Marcus, Mar 19 2019