A322483 The number of semi-unitary divisors of n.

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, 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, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

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) <= A000005(n) with equality if and only if n is squarefree (A005117).

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

Cf. A000005, A005117, A019554, A034444, A268335, A286324, A322482.

Cf. A001620, A056671, A343443, A365488, A365491.

Sequence in context: A098219 A368883 A173439 * A061389 A366763 A138011

Adjacent sequences: A322480 A322481 A322482 * A322484 A322485 A322486

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 11 2018

Status

approved