A322655 -id:A322655 - cp's OEIS frontend

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

1, 198, 608, 4680, 11322, 20826, 56608, 60192, 179424, 1737000, 2578968, 3055150, 3441888, 5604192, 6008184, 6331104, 302459850, 320457888, 477229032, 565344850, 579667086, 589459104, 731925000, 766073448, 907521650, 928765600, 3586977576, 3732082848, 6487717600

Offset: 1

Paolo P. Lava, Feb 22 2013

nonn

A066218 is a subsequence of this sequence.
Numbers k such that A000203(k) divides A007429(k). - Jaroslav Krizek, Dec 22 2018
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

			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.

Cf. A000203, A066218, A224488, A322655, A322656, A319296.

[k: k in [1..1000000] | &+[SumOfDivisors(d): d in Divisors(k)] mod  SumOfDivisors(k) eq 0] // Jaroslav Krizek, Dec 22 2018

with(numtheory);
A221219:=proc(q) local a,b,j,n;
for n from 1 to q do a:=divisors(n); b:=add(sigma(a[j]),j=1..nops(a));
  if type(b/sigma(n),integer) then print(n); fi; od; end:
A221219(10^10);

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 *)

isok(n) = (sumdiv(n, d, sigma(d)) % sigma(n) == 0); \\ Michel Marcus, Dec 22 2018

a(10)-a(28) from Donovan Johnson, Apr 05 2013

1 prepended by Jaroslav Krizek, Dec 22 2018

A322656 Denominator of (Sum_{d|n} sigma(d)) / sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 3, 8, 15, 13, 9, 12, 28, 14, 2, 24, 31, 18, 13, 20, 6, 32, 9, 24, 6, 31, 7, 20, 56, 30, 18, 32, 21, 48, 27, 16, 91, 38, 5, 56, 45, 42, 8, 44, 84, 13, 18, 48, 124, 19, 93, 72, 98, 54, 15, 72, 20, 16, 45, 60, 24, 62, 8, 52, 127, 4, 36, 68, 126, 96

Offset: 1

Jaroslav Krizek, Dec 22 2018

Denominator of A007429(n) / A000203(n).

Also denominator of Sum_{d|n} (sigma(d) / sigma(n)).

			For n = 4; a(4) = denominator((Sum_{d|4} sigma(d)) / sigma(4)) = denominator((1 + 3 + 7) / (1 + 2 + 4)) = denominator(11/7) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A007429, A319296, A221219, A322655 (numerator).

[Denominator(&+[SumOfDivisors(d): d in Divisors(n)] /  SumOfDivisors(n)): n in [1..1000]];

Table[Denominator[Sum[DivisorSigma[1, d], {d, Divisors[n]}] / DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 22 2018 *)

a(n) = denominator(sumdiv(n, d, sigma(d))/sigma(n)); \\ Michel Marcus, Dec 22 2018

a(n) = 1 for numbers in A221219.

a(n) = A000203(n) / gcd(A000203(n), A007429(n)). - Antti Karttunen, Nov 15 2021

cp's OEIS Frontend

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

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