A056626 Number of non-unitary square divisors of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset

1,32

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = A046951(n) - 2^r(n), where r(n) is the number of distinct prime factors of the largest unitary square divisor of n. [Corrected by Amiram Eldar, Aug 03 2024]

a(n) = A046951(n) - 2^(A162641(n)). - David A. Corneth, Jul 28 2017

From Amiram Eldar, Sep 26 2022: (Start)

a(n) = A046951(n) - A056624(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*(1 - 1/zeta(3)) = 0.27650128922802056073... . (End)

Example

n = p^u prime power has u+1 square divisors of which 2 (i.e., 1 and n) are unitary but u-1 are not unitary, so a(p^u) = u - 1. E.g., n = 4^4 = 256, has 5 square divisors {1, 4, 16, 64, 256} of which {4, 16, 64} are not unitary, so a(256)=3.

Mathematica

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, ! CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n])- Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Sep 26 2022 *)

Prog

(PARI) a(n) = {my(f = factor(n), r=0, m = 0); prod(i=1, #f~, f[i, 2]>>1 + 1) - 2^(omega(f) - omega(core(f)))} \\ David A. Corneth, Jul 28 2017
(PARI) a(n) = sumdiv(n, d, if(gcd(d, n/d)!=1, issquare(d))); \\ Michel Marcus, Jul 29 2017
(Python)
from math import prod
from sympy import factorint
def A056626(n):
    f = factorint(n).values()
    return prod((e>>1)+1 for e in f)-(1<<sum(e&1^1 for e in f)) # Chai Wah Wu, Aug 04 2024

Crossrefs

Cf. A000188, A008833, A034444, A046951, A055229, A056624, A162641.

Sequence in context: A104488 A244413 A318655 * A366123 A290081 A347706

Adjacent sequences: A056623 A056624 A056625 * A056627 A056628 A056629

Keyword

nonn,changed

Author

Labos Elemer, Aug 08 2000

Extensions

a(32) and a(96) corrected by Michael De Vlieger, Jul 29 2017

Status

approved