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
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Dec 04 2014
STATUS
approved