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A056624
Number of unitary square divisors of n.
21
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.
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
(Python)
from sympy import factorint
def A056624(n): return 1<<sum(e&1^1 for e in factorint(n).values()) # Chai Wah Wu, Aug 03 2024
KEYWORD
nonn,easy,mult
AUTHOR
Labos Elemer, Aug 08 2000
EXTENSIONS
More terms from Vladeta Jovovic, Dec 13 2002
STATUS
approved