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A034676 Sum of squares of unitary divisors of n. 5
1, 5, 10, 17, 26, 50, 50, 65, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 650, 626, 850, 730, 850, 842, 1300, 962, 1025, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also sum of unitary divisors of n^2. - Vladeta Jovovic, Nov 13 2001

If b(n,k)=sum of k-th powers of unitary divisors of n then b(n,k) is multiplicative with b(p^e,k)=p^(k*e)+1. - Vladeta Jovovic, Nov 13 2001

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unitary Divisor Function.

Wikipedia, Unitary divisor.

FORMULA

Multiplicative with a(p^e)=p^(2*e)+1.

Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2*s-2). - R. J. Mathar, Mar 04 2011

Sum_{k=1..n} a(k) ~ 30 * Zeta(3) * n^3 / Pi^4. - Vaclav Kotesovec, Jan 11 2019

Sum_{k>=1} 1/a(k) = 1.5594563610641446770272272038182777336348840179730233519185104374159616326... - Vaclav Kotesovec, Sep 20 2020

MATHEMATICA

f[n_] := Block[{d = Divisors@ n}, Plus @@ (Select[d, GCD[#, n/#] == 1 &]^2)]; Array[f, 50] (* Robert G. Wilson v, Mar 04 2011 *)

f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

PROG

(PARI) A034676_vec(len)={

        a000012=direuler(p=2, len, 1/(1-X)) ;

        a000290=direuler(p=2, len, 1/(1-p^2*X)) ;

        a000290x=direuler(p=2, len, 1-p^2*X^2) ;

        dirmul(dirmul(a000012, a000290), a000290x)

}

A034676_vec(70) ; /* via D.g.f., R. J. Mathar, Mar 05 2011 */

(Haskell)

a034676 = sum . map (^ 2) . a077610_row

-- Reinhard Zumkeller, Feb 12 2012

CROSSREFS

Cf. A034444, A034448, A034677, A034678-A034682.

Cf. A077610.

Sequence in context: A277186 A193053 A098749 * A076598 A306011 A080341

Adjacent sequences:  A034673 A034674 A034675 * A034677 A034678 A034679

KEYWORD

nonn,mult

AUTHOR

Erich Friedman

STATUS

approved

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Last modified November 23 21:51 EST 2020. Contains 338603 sequences. (Running on oeis4.)