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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.

%I A259924 #17 Sep 18 2024 20:35:33
%S A259924 1,780,1064,1289560,1428228,18107748,186000889725,680691912588
%N 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.
%C A259924 a(7) > 10^9. - _Giovanni Resta_, Jul 15 2015
%C A259924 a(9) > 10^13. - _Hiroaki Yamanouchi_, Sep 10 2015
%e A259924 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.
%e A259924 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.
%p A259924 with(numtheory): P:=proc(q) local a,k,n,p;
%p A259924 for n from 3 to q do a:=sort([op(divisors(n))]);
%p A259924 a:=add(a[k]*add(op(2,p)/op(1,p),p=ifactors(a[k])[2]),k=2..nops(a)-1);
%p A259924 if sigma(n)-n=a then print(n); fi; od; end: P(10^9);
%t A259924 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 *)
%Y A259924 Cf. A001065, A003415, A319683.
%K A259924 nonn,more
%O A259924 1,2
%A A259924 _Paolo P. Lava_, Jul 09 2015
%E A259924 a(6) from _Giovanni Resta_, Jul 15 2015
%E A259924 a(1) inserted and a(7)-a(8) added by _Hiroaki Yamanouchi_, Sep 10 2015