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

Author

Labos Elemer, Aug 08 2000

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)

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<Chai Wah Wu, Aug 04 2024

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)

Extensions

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