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%I #38 Sep 10 2023 02:02:31
%S 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,
%T 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,
%U 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
%N Number of unitary square divisors of n.
%C Unitary analog of A046951.
%H Antti Karttunen, <a href="/A056624/b056624.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) = 2^r, where r is the number of prime factors of the largest unitary square divisor of n.
%F Multiplicative with a(p^e) = 2^(1-(e mod 2)). - _Vladeta Jovovic_, Dec 13 2002
%F Dirichlet g.f.: zeta(s)*zeta(2*s)/zeta(3*s). - _Werner Schulte_, Apr 03 2018
%F 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
%F a(n) = 2^A162641(n). - _Amiram Eldar_, Sep 26 2022
%F a(n) = A034444(A350388(n)). - _Amiram Eldar_, Sep 09 2023
%e n=256, it has 5 square divisors of which only 2,{1,256} are unitary, 3 divisors are not.
%e n=124 has 2 (1 and 4) square divisors, both of them unitary a(124) = 2.
%e 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.
%t Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* _Michael De Vlieger_, Jul 28 2017 *)
%t f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 03 2022 *)
%o (Scheme) (define (A056624 n) (if (= 1 n) n (* (A000079 (A059841 (A067029 n))) (A056624 (A028234 n))))) ;; _Antti Karttunen_, Jul 28 2017
%o (PARI) a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, issquare(d))); \\ _Michel Marcus_, Jul 28 2017
%Y Cf. A000188, A008833, A034444, A046952, A055229, A056626, A059841, A162641, A350388.
%K nonn,easy,mult
%O 1,4
%A _Labos Elemer_, Aug 08 2000
%E More terms from _Vladeta Jovovic_, Dec 13 2002
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