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Number of divisors of n with an even sum of divisors.
1

%I #15 Oct 06 2023 10:53:49

%S 0,0,1,0,1,2,1,0,1,2,1,3,1,2,3,0,1,2,1,3,3,2,1,4,1,2,2,3,1,6,1,0,3,2,

%T 3,3,1,2,3,4,1,6,1,3,4,2,1,5,1,2,3,3,1,4,3,4,3,2,1,9,1,2,4,0,3,6,1,3,

%U 3,6,1,4,1,2,4,3,3,6,1,5,2,2,1,9,3,2,3,4,1,8,3,3

%N Number of divisors of n with an even sum of divisors.

%H Antti Karttunen, <a href="/A345221/b345221.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} ((sigma(d)+1) mod 2).

%F a(n) = A000005(n) - A065704(n). - _Antti Karttunen_, Dec 13 2021

%e a(24) = 4; The divisors d of 24 are {1, 2, 3, 4, 6, 8, 12, 24} with corresponding values of sigma(d) {1, 3, 4, 7, 12, 15, 28, 60}. There are 4 even values of sigma(d).

%t Table[Sum[Mod[DivisorSigma[1, k] + 1, 2] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

%t f1[p_, e_] := e+1; f2[p_, e_] := If[p == 2, e+1, Floor[e/2] + 1]; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Oct 06 2023 *)

%o (PARI) a(n) = sumdiv(n, d, !(sigma(d) % 2)); \\ _Michel Marcus_, Jun 11 2021

%Y Cf. A000005, A000203 (sigma), A065704.

%K nonn,easy

%O 1,6

%A _Wesley Ivan Hurt_, Jun 11 2021