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%I #43 Jan 22 2024 01:19:59
%S 1,2,2,2,2,4,2,3,2,4,2,4,2,4,4,3,2,4,2,4,4,4,2,6,2,4,3,4,2,8,2,4,4,4,
%T 4,4,2,4,4,6,2,8,2,4,4,4,2,6,2,4,4,4,2,6,4,6,4,4,2,8,2,4,4,4,4,8,2,4,
%U 4,8,2,6,2,4,4,4,4,8,2,6,3,4,2,8,4,4,4
%N The number of semi-unitary divisors of n.
%C The notion of semi-unitary divisor was introduced by Chidambaraswamy in 1967.
%C A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1. In terms of the relation defined in A322482, d is the largest divisor of n such that T(d, n/d) = 1 (the largest divisor d that is semiprime to n/d).
%C The number of divisors of n that are exponentially odd numbers (A268335). - _Amiram Eldar_, Sep 08 2023
%D J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.
%H Amiram Eldar, <a href="/A322483/b322483.txt">Table of n, a(n) for n = 1..10000</a>
%H Krishnaswami Alladi, <a href="https://doi.org/10.1017/S1446788700017304">On arithmetic functions and divisors of higher order</a>, Journal of the Australian Mathematical Society, Vol. 23, No. 1 (1977), pp. 9-27.
%H Pentti Haukkanen, <a href="https://www.researchgate.net/profile/Pentti_Haukkanen/publication/266363785_Basic_properties_of_the_bi-unitary_convolution_and_the_semi-unitary_convolution/links/5469efe50cf2397f782f4e3b/Basic-properties-of-the-bi-unitary-convolution-and-the-semi-unitary-convolution.pdf">Basic properties of the bi-unitary convolution and the semi-unitary convolution</a>, Indian J. Math, Vol. 40 (1998), pp. 305-315.
%H Vaclav Kotesovec, <a href="/A322483/a322483.jpg">Graph - the asymptotic ratio (100000 terms)</a>.
%H D. Suryanarayana, <a href="https://doi.org/10.1007/BFb0058797">The number of bi-unitary divisors of an integer</a>, The theory of arithmetic functions. Springer, Berlin, Heidelberg, 1972, pp. 273-282.
%H D. Suryanarayana and V. Siva Rama Prasad, <a href="https://doi.org/10.1017/S1446788700012908">Sum functions of k-ary and semi-k-ary divisors</a>, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
%H D. Suryanarayana and R. Sita Rama Chandra Rao, <a href="https://doi.org/10.1017/S1446788700016888">The number of unitarily k-free divisors of an integer</a>, Journal of the Australian Mathematical Society, Vol. 21, No. 1 (1976), pp. 19-35.
%H Laszlo Tóth, <a href="http://annalesm.elte.hu/annales41-1998/Annales_1998_T-XLI.pdf#page=167">Sum functions of certain generalized divisors</a>, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.
%F Multiplicative with a(p^e) = floor((e+3)/2).
%F a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
%F a(n) = Sum_{d|n} mu(d/gcd(d, n/d))^2. - _Ilya Gutkovskiy_, Feb 21 2020
%F a(n) = A000005(A019554(n)) (the number of divisors of the smallest number whose square is divisible by n). - _Amiram Eldar_, Sep 02 2023
%F From _Vaclav Kotesovec_, Sep 06 2023: (Start)
%F Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
%F Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
%F Let f(s) = Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
%F Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
%F f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444...,
%F f'(1) = f(1) * Sum_{p prime} (4*p-3) * log(p) / (p^3 - 2*p + 1) = 0.808661108949590913395... and gamma is the Euler-Mascheroni constant A001620. (End)
%e The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their number is 3, thus a(8) = 3.
%t f[p_, e_] := Floor[(e+3)/2]; sud[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[sud, 100]
%o (PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = (f[k,2]+3)\2; f[k,2] = 1;); factorback(f);} \\ _Michel Marcus_, Dec 14 2018
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1-X^2) * (1 + X - X^2))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 06 2023
%Y Cf. A000005, A005117, A019554, A034444, A268335, A286324, A322482.
%Y Cf. A001620, A056671, A343443, A365488, A365491.
%K nonn,easy,mult
%O 1,2
%A _Amiram Eldar_, Dec 11 2018
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