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A345320 Sum of the divisors of n whose square does not divide n.

%I A345320 #30 Jul 08 2025 00:31:12
%S A345320 0,2,3,4,5,11,7,12,9,17,11,25,13,23,23,24,17,35,19,39,31,35,23,57,25,
%T A345320 41,36,53,29,71,31,56,47,53,47,79,37,59,55,87,41,95,43,81,74,71,47,
%U A345320 117,49,87,71,95,53,116,71,117,79,89,59,165,61,95,100,112,83,143,67,123,95,143,71,183,73,113,118,137,95,167,79,179,108,125,83
%N A345320 Sum of the divisors of n whose square does not divide n.
%C A345320 Inverse Möbius transform of n+n^(1/2)*((-1)^tau(n)-1)/2. - _Wesley Ivan Hurt_, Jul 07 2025
%H A345320 Harvey P. Dale, <a href="/A345320/b345320.txt">Table of n, a(n) for n = 1..1000</a>
%F A345320 a(n) = Sum_{k=1..n} k * (ceiling(n/k^2) - floor(n/k^2)) * (1 - ceiling(n/k) + floor(n/k)).
%F A345320 a(n) = A000203(n) - A069290(n). - _Rémy Sigrist_, Jun 14 2021
%F A345320 a(n) = Sum_{d|n} (d+d^(1/2)*((-1)^tau(d)-1)/2). - _Wesley Ivan Hurt_, Jul 07 2025
%e A345320 a(16) = 24; The divisors of 16 whose square does not divide 16 are 8 and 16. The sum of the divisors is then 8 + 16 = 24.
%t A345320 Table[Sum[k (Ceiling[n/k^2] - Floor[n/k^2]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
%t A345320 sdnd[n_]:=Total[Select[Divisors[n],Mod[n,#^2]!=0&]]; Array[sdnd,100] (* _Harvey P. Dale_, Jul 07 2025 *)
%o A345320 (PARI) a(n) = sumdiv(n, d, if (n % d^2, d)); \\ _Michel Marcus_, Jun 13 2021
%o A345320 (Python)
%o A345320 from math import prod
%o A345320 from sympy import factorint
%o A345320 def A345320(n):
%o A345320     f = factorint(n).items()
%o A345320     return (prod(p**(q+1)-1 for p, q in f) - prod(p**(q//2+1)-1 for p, q in f))//prod(p-1 for p, q in f) # _Chai Wah Wu_, Jun 14 2021
%Y A345320 Cf. A000005 (tau), A000203 (sigma), A056595, A069290.
%K A345320 nonn
%O A345320 1,2
%A A345320 _Wesley Ivan Hurt_, Jun 13 2021