A035116 - OEIS

A035116

a(n) = tau(n)^2, where tau(n) = A000005(n).

1, 4, 4, 9, 4, 16, 4, 16, 9, 16, 4, 36, 4, 16, 16, 25, 4, 36, 4, 36, 16, 16, 4, 64, 9, 16, 16, 36, 4, 64, 4, 36, 16, 16, 16, 81, 4, 16, 16, 64, 4, 64, 4, 36, 36, 16, 4, 100, 9, 36, 16, 36, 4, 64, 16, 64, 16, 16, 4, 144, 4, 16

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OFFSET

1,2

REFERENCES

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 304.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.

FORMULA

Dirichlet g.f.: zeta(s)^4/zeta(2s).

tau(n)^2 = Sum_{d|n} tau(d^2), Dirichlet convolution of A048691 and A000012 (i.e.: inverse Mobius transform of A048691).

Multiplicative with a(p^e) = (e+1)^2. - Vladeta Jovovic, Dec 03 2001

G.f.: Sum_{n>=1} A000005(n^2)*x^n/(1-x^n). - Mircea Merca, Feb 25 2014

a(n) = A066446(n) + A184389(n). - Reinhard Zumkeller, Sep 08 2015

Let b(n), n > 0, be the Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(3,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^4/(zeta(2*s))^3. - Werner Schulte, Feb 07 2021

MAPLE

A035116 := proc(n) numtheory[tau](n)^2 ; end proc:

seq(A035116(n), n=1..40) ; # R. J. Mathar, Apr 02 2011

MATHEMATICA

DivisorSigma[0, Range[100]]^2 (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

PROG

(Magma) [ NumberOfDivisors(n)^2 : n in [1..100] ];

(PARI) A035116(n)=numdiv(n)^2;

(Haskell)

a035116 = (^ 2) . a000005' -- Reinhard Zumkeller, Sep 08 2015

CROSSREFS

Cf. A000005, A048691, A061391.