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

1,1

Comments

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000203, A007955, A187680.

Includes A188061.

Sequence in context: A312992 A312993 A312994 * A312995 A312996 A312997

Adjacent sequences: A346953 A346954 A346955 * A346957 A346958 A346959

Keyword

nonn

Author

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

Status

approved