A249912 Numbers whose sum of anti-divisors is equal to the sum of the divisors of their arithmetic derivative.

26, 51, 134, 369, 614, 1154, 2010, 2186, 2790, 3134, 4034, 5294, 6074, 6614, 7814, 9134, 11031, 12014, 12494, 13158, 15014, 22394, 22934, 22994, 24554, 27134, 32894, 47774, 52694, 54794, 62714, 65414, 75494, 87194, 101054, 112754, 114974, 126974, 150074, 156014

Offset

1,1

Comments

sigma*(n) = sigma(n’), where sigma*(n) is the sum of anti-divisors and n’ is the arithmetic derivative of n.

Majority of the terms end in 4.

Links

Paolo P. Lava, Table of n, a(n) for n = 1..100

Example

The anti-divisors of 26 are 3, 4, 17 and their sum is 24; arithmetic derivative of 26 is 15 and sigma(15) = 24.

Maple

with(numtheory): P:=proc(q) local a, i, j, k, n, p;
for n from 1 to q do i:=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if a=sigma(i) then print(n); fi; od; end: P(10^6);

Prog

(PARI) isok(n) = my(k=valuation(n, 2)); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2 == sigma(sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i])); \\ Michel Marcus, Dec 06 2014

Crossrefs

Cf. A000203 (sum of divisors), A003415 (arithmetic derivative).

Cf. A066417 (sum of anti-divisors), A229342 (sum of divisors of arithmetic derivative).

Sequence in context: A044459 A158060 A169861 * A267039 A122999 A040650

Adjacent sequences: A249909 A249910 A249911 * A249913 A249914 A249915

Keyword

nonn

Author

Paolo P. Lava, Dec 04 2014

Status

approved