A188061 -id:A188061 - cp's OEIS frontend

A187680 a(n) = (product of divisors of n) mod (sum of divisors of n).

0, 2, 3, 1, 5, 0, 7, 4, 1, 10, 11, 20, 13, 4, 9, 1, 17, 21, 19, 20, 25, 16, 23, 36, 1, 4, 9, 0, 29, 0, 31, 8, 33, 22, 25, 83, 37, 4, 9, 40, 41, 48, 43, 8, 21, 28, 47, 88, 1, 8, 9, 76, 53, 96, 1, 16, 49, 34, 59, 120, 61, 4, 31, 1

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2011

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A007955, A145551, A188061.

A187680 := proc(n) A007955(n) mod numtheory[sigma](n) ; end proc:
seq(A187680(n),n=1..120) ; # R. J. Mathar, Mar 17 2011

Table[Mod[Times@@Divisors[n],DivisorSigma[1,n]],{n,70}] (* Harvey P. Dale, May 23 2021 *)
a[n_] := Mod[n^(DivisorSigma[0, n]/2), DivisorSigma[1, n]]; Array[a, 60] (* Amiram Eldar, Jun 18 2022 *)

a(n) = my(d=divisors(n)); vecprod(d) % vecsum(d); \\ Michel Marcus, Jan 29 2019

a(n) = A007955(n) mod A000203(n).

a(n) = 0 iff n is in A145551 and a(n) = 1 iff n is in A188061. - Amiram Eldar, Jun 18 2022

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

Original entry on oeis.org

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

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

nonn

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

			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.

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

Cf. A000203, A007955, A187680.

Includes A188061.

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]);

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 *)

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

cp's OEIS Frontend

A187680 a(n) = (product of divisors of n) mod (sum of divisors of n).

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