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A221219
Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).
6
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
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
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