OFFSET
1,2
COMMENTS
The notion of semi-unitary divisor was introduced by Chidambaraswamy in 1967.
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).
The number of divisors of n that are exponentially odd numbers (A268335). - Amiram Eldar, Sep 08 2023
REFERENCES
J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, Journal of the Australian Mathematical Society, Vol. 23, No. 1 (1977), pp. 9-27.
Pentti Haukkanen, Basic properties of the bi-unitary convolution and the semi-unitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305-315.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions. Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of unitarily k-free divisors of an integer, Journal of the Australian Mathematical Society, Vol. 21, No. 1 (1976), pp. 19-35.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.
FORMULA
Multiplicative with a(p^e) = floor((e+3)/2).
a(n) = Sum_{d|n} mu(d/gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
a(n) = A000005(A019554(n)) (the number of divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 02 2023
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Let f(s) = Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
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(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444...,
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)
EXAMPLE
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.
MATHEMATICA
f[p_, e_] := Floor[(e+3)/2]; sud[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[sud, 100]
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 11 2018
STATUS
approved