A065960 a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).

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

Offset: 1

Author

N. J. A. Sloane, Dec 08 2001

Keywords

• nonn
• mult
• easy

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.

Crossrefs

Cf. A000010, A007434, A059377, A069093.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), this sequence (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

Programs

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 *)

PARI

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

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)