login
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.
0
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
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
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