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

1,2

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

Links

Table of n, a(n) for n=1..29.

Example

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.

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 *)

Prog

(Magma) [k: k in [1..1000000] | &+[SumOfDivisors(d): d in Divisors(k)] mod  SumOfDivisors(k) eq 0] // Jaroslav Krizek, Dec 22 2018
(PARI) isok(n) = (sumdiv(n, d, sigma(d)) % sigma(n) == 0); \\ Michel Marcus, Dec 22 2018

Crossrefs

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

Sequence in context: A075293 A083264 A202526 * A238765 A066218 A304614

Adjacent sequences: A221216 A221217 A221218 * A221220 A221221 A221222

Keyword

nonn

Author

Paolo P. Lava, Feb 22 2013

Extensions

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

1 prepended by Jaroslav Krizek, Dec 22 2018

Status

approved