A065958 a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).

1, 5, 10, 20, 26, 50, 50, 80, 90, 130, 122, 200, 170, 250, 260, 320, 290, 450, 362, 520, 500, 610, 530, 800, 650, 850, 810, 1000, 842, 1300, 962, 1280, 1220, 1450, 1300, 1800, 1370, 1810, 1700, 2080, 1682, 2500, 1850, 2440, 2340, 2650, 2210

Offset: 1

Author

N. J. A. Sloane, Dec 08 2001

Keywords

• nonn
• mult
• easy

Comments

The sequence may be considered as psi_2, a generalization of Dedekind psi function, where psi_1 is A001615. - Enrique Pérez Herrero, Jul 06 2011

References

• József Sándor, Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press, Rehoboth 2002, pp. 193.

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.
• R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 Chapter 3.14.2

Crossrefs

Cf. A000010, A007434, A156733, A301978, A301980, A321973.

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), this sequence (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

Programs

Maple

A065958 := proc(n) local i,j,k,t1,t2,t3; t1 := ifactors(n)[2]; t2 := n^2*mul((1+1/(t1[i][1])^2),i=1..nops(t1)); end;

Mathematica

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A065958[n_]:=JordanTotient[n,4]/JordanTotient[n,2]; (* Enrique Pérez Herrero, Aug 22 2010 *)
f[p_, e_] := p^(2*e) + p^(2*(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*sumdiv(n,d,moebius(d)^2/d^2),","))

PARI

a(n)=sumdiv(n,d,moebius(n/d)^2*d^2); /* Joerg Arndt, Jul 06 2011 */

Formula

Multiplicative with a(p^e) = p^(2*e) + p^(2*e-2). - Vladeta Jovovic, Dec 09 2001

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

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

Inverse Euler transform of n*A156733(n). - Paul D. Hanna and Vladeta Jovovic, Feb 14 2009

From Enrique Pérez Herrero, Aug 22 2010: (Start)

a(n) = J_4(n)/(phi(n)*psi(n)) = A059377(n)/(A001615(n)*A000010(n))

a(n) = J_4(n)/J_2(n) = A059377(n)/A007434(n), where J_k is the k-th Jordan totient function. (End)

Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s). Dirichlet convolution of A008966 and A000290. - R. J. Mathar, Apr 10 2011

G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^4 - 1)) = 1.5421162831401587416523241690601522041445615542162573163112157073779258386... - Vaclav Kotesovec, Sep 19 2020

a(n) = Sum_{d|n} d*phi(d)*psi(n/d). - Ridouane Oudra, Jan 01 2021

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = Sum_{k=1..n} psi(gcd(n,k))*n/gcd(n,k), where psi(n) = A001615(n).

a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Sum_{k=1..n} a(k) ~ c * n^3, where c = 315*zeta(3)/Pi^6 = 0.393854... . - Amiram Eldar, Oct 19 2022