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

Author

Paolo P. Lava, Jul 09 2015

Keywords

• nonn
• more

Comments

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

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

Examples

			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.
		

Crossrefs

Cf. A001065, A003415, A319683.

Programs

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 *)

Extensions

a(6) from Giovanni Resta, Jul 15 2015

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