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URL: https://oeis.org/A065958

⇱ A065958 - OEIS

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A065958
a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).
22
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
1,2
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
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
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
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 *)
PROG
(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 */
CROSSREFS
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).
Sequence in context: A072703 A086761 A045191 * A065969 A306775 A027884
KEYWORD
nonn,mult,easy
AUTHOR
N. J. A. Sloane, Dec 08 2001
STATUS
approved