A346956 Numbers k such that A000203(k) and A007955(k) are both divisible by A187680(k).

4, 9, 14, 16, 25, 38, 42, 49, 51, 55, 62, 64, 70, 81, 86, 92, 96, 117, 121, 130, 134, 138, 140, 158, 159, 161, 168, 169, 182, 206, 209, 234, 254, 256, 266, 267, 278, 282, 284, 289, 302, 322, 326, 351, 361, 376, 390, 398, 408, 410, 422, 426, 434, 446, 477, 508, 529, 532, 534, 542, 551, 566, 590

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 08 2021

Keywords

• nonn

Comments

Numbers k such that both the sum s and product p of the divisors of k are divisible by (p mod s).

Examples

			a(3) = 14 is a term because A000203(14) = 1+2+7+14 = 24, A007955(14) = 1*2*7*14 = 196, A187680(14) = 196 mod 24 = 4, and both 24 and 196 are divisible by 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A007955, A187680.

Includes A188061.

Programs

Maple

filter:= proc(n) local d,s,p,r;
  d:= numtheory:-divisors(n);
  s:= convert(d,`+`);
  p:= convert(d,`*`);
  r:= p mod s;
  r <> 0 and p mod r = 0 and s mod r = 0
end proc:
select(filter, [$1..1000]);

Mathematica

okQ[n_] := Module[{d, s, p, m},
  d = Divisors[n];
  s = Total[d];
  p = Times @@ d;
  m = Mod[p, s];
  If[m == 0, False, Divisible[s, m] && Divisible[p, m]]];
Select[Range[1000], okQ] (* Jean-François Alcover, May 16 2023 *)

PARI

isok(k) = my(d=divisors(k), s=vecsum(d), p=vecprod(d), m=p % s); (m>0) && !(s%m) && !(p%m); \\ Michel Marcus, Aug 09 2021