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A320059 Sum of divisors of n^2 that do not divide n.

%I A320059 #25 Sep 08 2022 08:46:23
%S A320059 0,4,9,24,25,79,49,112,108,199,121,375,169,375,379,480,289,808,361,
%T A320059 919,709,895,529,1591,750,1239,1053,1711,841,2749,961,1984,1681,2095,
%U A320059 1719,3660,1369,2607,2323,3847,1681,5091,1849,4039,3673,3799,2209,6519,2744,5374
%N A320059 Sum of divisors of n^2 that do not divide n.
%C A320059 sigma(n^2) is always odd, so this sequence has the opposite parity from sigma(n): even if n is a square or twice a square, odd otherwise.
%H A320059 Chai Wah Wu, <a href="/A320059/b320059.txt">Table of n, a(n) for n = 1..10000</a>
%F A320059 a(n) = sigma(n^2) - sigma(n).
%F A320059 a(n) = A065764(n) - A000203(n).
%F A320059 a(n) = n^2 iff n is prime. - _Altug Alkan_, Oct 04 2018
%p A320059 map(n -> numtheory:-sigma(n^2)-numtheory:-sigma(n), [$1..100]); # _Robert Israel_, Oct 04 2018
%t A320059 Table[DivisorSigma[1, n^2] - DivisorSigma[1, n], {n, 70}] (* _Vincenzo Librandi_, Oct 05 2018 *)
%o A320059 (PARI) a(n) = sigma(n^2)-sigma(n)
%o A320059 (Magma) [DivisorSigma(1, n^2) - DivisorSigma(1, n): n in [1..70]]; // _Vincenzo Librandi_, Oct 05 2018
%o A320059 (Python)
%o A320059 from __future__ import division
%o A320059 from sympy import factorint
%o A320059 def A320059(n):
%o A320059     c1, c2 = 1, 1
%o A320059     for p, a in factorint(n).items():
%o A320059         c1 *= (p**(2*a+1)-1)//(p-1)
%o A320059         c2 *= (p**(a+1)-1)//(p-1)
%o A320059     return c1-c2 # _Chai Wah Wu_, Oct 05 2018
%Y A320059 Cf. A000203, A065764, A072861, A054785.
%K A320059 nonn,look
%O A320059 1,2
%A A320059 _Franklin T. Adams-Watters_, Oct 04 2018