A071327 -id:A071327 - cp's OEIS frontend

A071326 Sum of squares > 1 dividing n.

0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 4, 0, 0, 0, 20, 0, 9, 0, 4, 0, 0, 0, 4, 25, 0, 9, 4, 0, 0, 0, 20, 0, 0, 0, 49, 0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 20, 49, 25, 0, 4, 0, 9, 0, 4, 0, 0, 0, 4, 0, 0, 9, 84, 0, 0, 0, 4, 0, 0, 0, 49, 0, 0, 25, 4, 0, 0, 0, 20, 90, 0, 0, 4, 0, 0, 0, 4, 0, 9, 0, 4, 0, 0, 0, 20, 0, 49, 9, 129

Offset: 1

Reinhard Zumkeller, May 18 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A010052, A071325, A071327.

One less than A035316.

Array[DivisorSum[#, # &, IntegerQ@ Sqrt@ # &] - 1 &, 100] (* Michael De Vlieger, Nov 17 2017 *)
f[p_, e_] := (p^(2*(1+Floor[e/2]))-1)/(p^2-1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - 1; Array[a, 100] (* Amiram Eldar, May 15 2025 *)

A071326(n) = sumdiv(n,d,(d>1)*issquare(d)*d); \\ Antti Karttunen, Nov 17 2017

a(n) = A035316(n)-1.

Sum_{d|n, d>1} A010052(d). - Antti Karttunen, Nov 17 2017

More terms from Antti Karttunen, Nov 17 2017

A370239 The sum of divisors of n that are squares of squarefree numbers.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 5, 10, 1, 1

Offset: 1

Amiram Eldar, Feb 13 2024

The number of these divisors is A323308(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A036785, A048250, A062503, A071327, A078434, A323308, A370240.

f[p_, e_] := If[e == 1, 1, 1 + p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + f[i,1]^2));}

Multiplicative with a(p) = 1 and a(p^e) = 1 + p^2 for e >= 2.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) = A071327(n) + 1 if and only if n is not in A036785.
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(4*s-4).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 2*zeta(3/2)/Pi^2 = 0.5293779248... .

cp's OEIS Frontend

A071326 Sum of squares > 1 dividing n.

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