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A221219 Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).

%I A221219 #47 Sep 08 2022 08:46:04
%S A221219 1,198,608,4680,11322,20826,56608,60192,179424,1737000,2578968,
%T A221219 3055150,3441888,5604192,6008184,6331104,302459850,320457888,
%U A221219 477229032,565344850,579667086,589459104,731925000,766073448,907521650,928765600,3586977576,3732082848,6487717600
%N A221219 Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).
%C A221219 A066218 is a subsequence of this sequence.
%C A221219 Numbers k such that A000203(k) divides A007429(k). - _Jaroslav Krizek_, Dec 22 2018
%C A221219 Corresponding values of (Sum_{d|k} sigma(d)) / sigma(k)  for numbers k from this sequence: 1, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, ... - _Jaroslav Krizek_, Dec 22 2018
%e A221219 4680 is in the sequence because sigma(4680)=16380, its proper divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24, 26, 30, 36, 39, 40, 45, 52, 60, 65, 72, 78, 90, 104, 117, 120, 130, 156, 180, 195, 234, 260, 312, 360, 390, 468, 520, 585, 780, 936, 1170, 1560, 2340 and the sum of their sigma values is 32760. Finally 32760/16380=2.
%p A221219 with(numtheory);
%p A221219 A221219:=proc(q) local a,b,j,n;
%p A221219 for n from 1 to q do a:=divisors(n); b:=add(sigma(a[j]),j=1..nops(a));
%p A221219   if type(b/sigma(n),integer) then print(n); fi; od; end:
%p A221219 A221219(10^10);
%t A221219 f1[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f2[p_, e_] := (p^(e+1) - 1)/(p - 1); aQ[1] = True; aQ[n_] := Module[{f = FactorInteger[n]}, Divisible[Times @@ f1 @@@ f, Times @@ f2 @@@ f]]; Select[Range[10^5], aQ] (* _Amiram Eldar_, Dec 23 2018 *)
%o A221219 (Magma) [k: k in [1..1000000] | &+[SumOfDivisors(d): d in Divisors(k)] mod  SumOfDivisors(k) eq 0] // _Jaroslav Krizek_, Dec 22 2018
%o A221219 (PARI) isok(n) = (sumdiv(n, d, sigma(d)) % sigma(n) == 0); \\ _Michel Marcus_, Dec 22 2018
%Y A221219 Cf. A000203, A066218, A224488, A322655, A322656, A319296.
%K A221219 nonn
%O A221219 1,2
%A A221219 _Paolo P. Lava_, Feb 22 2013
%E A221219 a(10)-a(28) from _Donovan Johnson_, Apr 05 2013
%E A221219 1 prepended by _Jaroslav Krizek_, Dec 22 2018