A249912 - OEIS

%I #30 Dec 09 2014 08:15:55

%S 26,51,134,369,614,1154,2010,2186,2790,3134,4034,5294,6074,6614,7814,

%T 9134,11031,12014,12494,13158,15014,22394,22934,22994,24554,27134,

%U 32894,47774,52694,54794,62714,65414,75494,87194,101054,112754,114974,126974,150074,156014

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

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

%C Majority of the terms end in 4.

%H Paolo P. Lava, <a href="/A249912/b249912.txt">Table of n, a(n) for n = 1..100</a>

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

%p with(numtheory): P:=proc(q) local a,i,j,k,n,p;

%p for n from 1 to q do i:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);

%p k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;

%p a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;

%p if a=sigma(i) then print(n); fi; od; end: P(10^6);

%o (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

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

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

%K nonn

%O 1,1

%A _Paolo P. Lava_, Dec 04 2014