A065959 a(n) = n^3*Product_{distinct primes p dividing n} (1+1/p^3).

1, 9, 28, 72, 126, 252, 344, 576, 756, 1134, 1332, 2016, 2198, 3096, 3528, 4608, 4914, 6804, 6860, 9072, 9632, 11988, 12168, 16128, 15750, 19782, 20412, 24768, 24390, 31752, 29792, 36864, 37296, 44226, 43344, 54432, 50654, 61740, 61544

Offset

1,2

Links

Enrique 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^(3*e)+p^(3*e-3). - Vladeta Jovovic, Dec 09 2001

a(n) = n^3*Sum_{d|n} mu(d)^2/d^3. - Benoit Cloitre, Apr 07 2002

a(n) = Sum_{d|n} mu(n/d)^2*d^3. - Joerg Arndt, Jul 06 2011

a(n) = J_6(n)/J_3(n) = A069091(n)/A059376(n). - Enrique Pérez Herrero, Aug 22 2010

Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2*s). Dirichlet convolution of A008966 and A000578. - R. J. Mathar, Apr 10 2011

G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + 4*x^k + x^(2*k))/(1 - x^k)^4. - Ilya Gutkovskiy, Oct 24 2018

From Vaclav Kotesovec, Sep 19 2020: (Start)

Sum_{k=1..n} a(k) ~ 105*n^4 / (4*Pi^4).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/(p^6-1)) = 1.18370753651668075930203278269930233284040397061087910806697928843547863257... (End)

Mathematica

JordanTotient[n_, k_:1] := DivisorSum[n, #^k * MoebiusMu[n/#] &]/; (n>0) && IntegerQ[n]; A065959[n_] := JordanTotient[n, 6] / JordanTotient[n, 3]; Array[A065959, 39] (* Enrique Pérez Herrero, Aug 22 2010 *)
f[p_, e_] := p^(3*e) + p^(3*(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^3*sumdiv(n, d, moebius(d)^2/d^3), ", "))
(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^3); \\ Joerg Arndt, Jul 06 2011

Crossrefs

Cf. A000010, A007434.

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), this sequence (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

Sequence in context: A034677 A009255 A062451 * A226333 A017669 A277065

Adjacent sequences: A065956 A065957 A065958 * A065960 A065961 A065962

Keyword

nonn,mult,easy

Author

N. J. A. Sloane, Dec 08 2001

Status

approved