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A322485 The sum of the semi-unitary divisors of n. 4
1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 19, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 76, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 71, 84, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 (see A322483).

REFERENCES

J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

LINKS

Table of n, a(n) for n=1..66.

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 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) = sigma(p^floor((e-1)/2)) + p^e = (p^floor((e+1)/2) - 1)/(p-1) + p^e.

In particular a(p) = p + 1, a(p^2) = p^2 + 1, a(p^3) = p^3 + p + 1.

a(n) <= A000203(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 sum is 11, thus a(8) = 11.

MATHEMATICA

f[p_, e_] := (p^Floor[(e+1)/2] - 1)/(p-1) + p^e; susigma[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[susigma, 100]

PROG

(PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k, 1], e=f[k, 2]); f[k, 1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k, 2] = 1; ); factorback(f); } \\ Michel Marcus, Dec 14 2018

CROSSREFS

Cf. A000203, A005117, A034448, A188999, A322483.

Sequence in context: A069184 A181549 A241405 * A324706 A049417 A188999

Adjacent sequences:  A322482 A322483 A322484 * A322486 A322487 A322488

KEYWORD

nonn,mult

AUTHOR

Amiram Eldar, Dec 11 2018

STATUS

approved

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Last modified July 21 22:02 EDT 2019. Contains 325210 sequences. (Running on oeis4.)