A056624 Number of unitary square divisors of n.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1

Offset

1,4

Comments

Unitary analog of A046951.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

Formula

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

Multiplicative with a(p^e) = 2^(1-(e mod 2)). - Vladeta Jovovic, Dec 13 2002

Dirichlet g.f.: zeta(s)*zeta(2*s)/zeta(3*s). - Werner Schulte, Apr 03 2018

Sum_{k=1..n} a(k) ~ n*Pi^2/(6*zeta(3)) + sqrt(n)*zeta(1/2)/zeta(3/2). - Vaclav Kotesovec, Feb 07 2019

a(n) = 2^A162641(n). - Amiram Eldar, Sep 26 2022

a(n) = A034444(A350388(n)). - Amiram Eldar, Sep 09 2023

Example

n=256, it has 5 square divisors of which only 2,{1,256} are unitary, 3 divisors are not.
n=124 has 2 (1 and 4) square divisors, both of them unitary a(124) = 2.
n=108 has 12 divisors, 4 square divisors: {1,4,9,36} of which 1 and 4 are unitary, 9 and 36 are not. So a(108)=2. The largest unitary square divisor of 108 is 4 with 1 prime divisor so a(108) = 2^1 = 2.

Mathematica

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2022 *)

Prog

(Scheme) (define (A056624 n) (if (= 1 n) n (* (A000079 (A059841 (A067029 n))) (A056624 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017
(PARI) a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, issquare(d))); \\ Michel Marcus, Jul 28 2017

Crossrefs

Cf. A000188, A008833, A034444, A046952, A055229, A056626, A059841, A162641, A350388.

Sequence in context: A368885 A294932 A327804 * A193348 A263723 A354974

Adjacent sequences: A056621 A056622 A056623 * A056625 A056626 A056627

Keyword

nonn,easy,mult

Author

Labos Elemer, Aug 08 2000

Extensions

More terms from Vladeta Jovovic, Dec 13 2002

Status

approved