

A322485


The sum of the semiunitary 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 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 (see A322483).


REFERENCES

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


LINKS

Table of n, a(n) for n=1..66.
Pentti Haukkanen, Basic properties of the biunitary convolution and the semiunitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305315.
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) = sigma(p^floor((e1)/2)) + p^e = (p^floor((e+1)/2)  1)/(p1) + 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 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 sum is 11, thus a(8) = 11.


MATHEMATICA

f[p_, e_] := (p^Floor[(e+1)/2]  1)/(p1) + 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)/(p1) + 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



