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A322483 The number of semi-unitary 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 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 semi-prime to n/d).

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.

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.

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).

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

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

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Last modified October 19 23:44 EDT 2019. Contains 328244 sequences. (Running on oeis4.)