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A249912 Numbers whose sum of anti-divisors is equal to the sum of the divisors of their arithmetic derivative. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 24 19:34 EST 2022. Contains 350565 sequences. (Running on oeis4.)