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

Author

Paolo P. Lava, Feb 22 2013

Keywords

• nonn

Comments

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

Examples

			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.
		

Crossrefs

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

Programs

Magma

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

Maple

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);

Mathematica

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 *)

PARI

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

Extensions

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

1 prepended by Jaroslav Krizek, Dec 22 2018