A259924 Numbers n such that sigma(n) - n = sum_{k divides n, k < n} k', where sigma(n) is the sum of the divisors of n and k' is the arithmetic derivative of k.

1, 780, 1064, 1289560, 1428228, 18107748, 186000889725, 680691912588

Offset

1,2

Comments

a(7) > 10^9. - Giovanni Resta, Jul 15 2015

a(9) > 10^13. - Hiroaki Yamanouchi, Sep 10 2015

Links

Table of n, a(n) for n=1..8.

Example

Aliquot parts of 780 are 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390. Their arithmetic derivatives are 0, 1, 1, 4, 1, 5, 7, 16, 1, 8, 24, 15, 31, 16, 56, 92, 18, 71, 101, 220, 119, 332, 433. Their sum is 1572 and sigma(780) - 780 = 2352 - 780 = 1572.
Aliquot parts of 1064 are 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532. Their arithmetic derivatives are 0, 1, 4, 1, 12, 9, 1, 32, 21, 92, 80, 26, 236, 185, 636. Their sum is 1336 and sigma(1064) - 1064 = 2400 - 1064 = 1336.

Maple

with(numtheory): P:=proc(q) local a, k, n, p;
for n from 3 to q do a:=sort([op(divisors(n))]);
a:=add(a[k]*add(op(2, p)/op(1, p), p=ifactors(a[k])[2]), k=2..nops(a)-1);
if sigma(n)-n=a then print(n); fi; od; end: P(10^9);

Mathematica

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; Select[Range@ 1500000, DivisorSigma[1, #] - # == Total[f /@ Most@ Divisors@ #] &] (* Michael De Vlieger, Jul 16 2015, after Michael Somos at A003415 *)

Crossrefs

Cf. A001065, A003415.

Sequence in context: A231252 A341190 A147547 * A135198 A292063 A250952

Adjacent sequences: A259921 A259922 A259923 * A259925 A259926 A259927

Keyword

nonn,more

Author

Paolo P. Lava, Jul 09 2015

Extensions

a(6) from Giovanni Resta, Jul 15 2015

a(1) inserted and a(7)-a(8) added by Hiroaki Yamanouchi, Sep 10 2015

Status

approved