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

%I A346956 #16 May 16 2023 10:40:04
%S A346956 4,9,14,16,25,38,42,49,51,55,62,64,70,81,86,92,96,117,121,130,134,138,
%T A346956 140,158,159,161,168,169,182,206,209,234,254,256,266,267,278,282,284,
%U A346956 289,302,322,326,351,361,376,390,398,408,410,422,426,434,446,477,508,529,532,534,542,551,566,590
%N A346956 Numbers k such that A000203(k) and A007955(k) are both divisible by A187680(k).
%C A346956 Numbers k such that both the sum s and product p of the divisors of k are divisible by (p mod s).
%H A346956 Robert Israel, <a href="/A346956/b346956.txt">Table of n, a(n) for n = 1..10000</a>
%e A346956 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.
%p A346956 filter:= proc(n) local d,s,p,r;
%p A346956   d:= numtheory:-divisors(n);
%p A346956   s:= convert(d,`+`);
%p A346956   p:= convert(d,`*`);
%p A346956   r:= p mod s;
%p A346956   r <> 0 and p mod r = 0 and s mod r = 0
%p A346956 end proc:
%p A346956 select(filter, [$1..1000]);
%t A346956 okQ[n_] := Module[{d, s, p, m},
%t A346956   d = Divisors[n];
%t A346956   s = Total[d];
%t A346956   p = Times @@ d;
%t A346956   m = Mod[p, s];
%t A346956   If[m == 0, False, Divisible[s, m] && Divisible[p, m]]];
%t A346956 Select[Range[1000], okQ] (* _Jean-François Alcover_, May 16 2023 *)
%o A346956 (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
%Y A346956 Cf. A000203, A007955, A187680.
%Y A346956 Includes A188061.
%K A346956 nonn
%O A346956 1,1
%A A346956 _J. M. Bergot_ and _Robert Israel_, Aug 08 2021