

A056626


Number of nonunitary square divisors of n.


7



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
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OFFSET

1,32


LINKS

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


FORMULA

a(n) = A046951(n)  2^r, where r is the number of prime factors in the largest unitary prime divisor of n.
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 u1 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


CROSSREFS

Cf. A000188, A008833, A034444, A046951, A055229, A056624, A162641.
Sequence in context: A104488 A244413 A318655 * A290081 A347706 A348381
Adjacent sequences: A056623 A056624 A056625 * A056627 A056628 A056629


KEYWORD

nonn


AUTHOR

Labos Elemer, Aug 08 2000


EXTENSIONS

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


STATUS

approved



