

A322483


The number of semiunitary divisors of n.


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The notion of semiunitary divisor was introduced by Chidambaraswamy in 1967.
A semiunitary 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).


REFERENCES

J. Chidambaraswamy, Sum functions of unitary and semiunitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117126.


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. 927.
Pentti Haukkanen, Basic properties of the biunitary convolution and the semiunitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305315.
D. Suryanarayana, The number of biunitary divisors of an integer, The theory of arithmetic functions. Springer, Berlin, Heidelberg, 1972, pp. 273282.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of kary and semikary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148162.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165180.


FORMULA

Multiplicative with a(p^e) = floor((e+3)/2).
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).


EXAMPLE

The semiunitary divisors of 8 are 1, 2, 8 (4 is not semiunitary 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


CROSSREFS

Cf. A000005, A005117, A034444, A286324, A322482.
Sequence in context: A048003 A098219 A173439 * A061389 A138011 A036555
Adjacent sequences: A322480 A322481 A322482 * A322484 A322485 A322486


KEYWORD

nonn,mult


AUTHOR

Amiram Eldar, Dec 11 2018


STATUS

approved



