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A035116 a(n) = tau(n)^2, where tau(n) = A000005(n). 30
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A345732 A023405 A160900 * A088613 A351582 A049723
KEYWORD
nonn,easy,mult
AUTHOR
EXTENSIONS
Additional comments from Vladeta Jovovic, Apr 29 2001
STATUS
approved

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Last modified February 21 02:30 EST 2024. Contains 370219 sequences. (Running on oeis4.)