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A065960 a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4). 5
1, 17, 82, 272, 626, 1394, 2402, 4352, 6642, 10642, 14642, 22304, 28562, 40834, 51332, 69632, 83522, 112914, 130322, 170272, 196964, 248914, 279842, 356864, 391250, 485554, 538002, 653344, 707282, 872644, 923522, 1114112, 1200644 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

E. Pérez Herrero, Table of n, a(n) for n=1..10000

F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.

Wikipedia, Dedekind Psi function.

FORMULA

Multiplicative with a(p^e) = p^(4*e)+p^(4*e-4). - Vladeta Jovovic, Dec 09 2001

a(n) = n^4 * Sum_{d|n} mu(d)^2/d^4. - Benoit Cloitre, Apr 07 2002

a(n) = J_8(n)/J_4(n) = A069093(n)/A059377(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Aug 29 2010

Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(2*s). - R. J. Mathar, Jun 06 2011

From Vaclav Kotesovec, Sep 19 2020: (Start)

Sum_{k=1..n} a(k) ~ 18711*zeta(5)*n^5 / Pi^10.

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/(p^8-1)) = 1.078178802583045599985995264729541574821218371712364313741065126120993131... (End)

MAPLE

A065960 := proc(n) n^4*mul(1+1/p^4, p=numtheory[factorset](n)) ; end proc:

seq(A065960(n), n=1..20) ; # R. J. Mathar, Jun 06 2011

MATHEMATICA

a[n_] := n^4*DivisorSum[n, MoebiusMu[#]^2/#^4&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)

f[p_, e_] := p^(4*e) + p^(4*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PROG

(PARI) for(n=1, 100, print1(n^4*sumdiv(n, d, moebius(d)^2/d^4), ", "))

CROSSREFS

Cf. A000010, A001615, A007434, A059377, A065959, A065958, A069093.

Sequence in context: A088687 A321560 A034678 * A017671 A001159 A053820

Adjacent sequences:  A065957 A065958 A065959 * A065961 A065962 A065963

KEYWORD

nonn,mult,easy

AUTHOR

N. J. A. Sloane, Dec 08 2001

STATUS

approved

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Last modified November 27 12:51 EST 2021. Contains 349394 sequences. (Running on oeis4.)