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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 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. Cf. A066446, A184389, A061502. Sequence in context: A345732 A023405 A160900 * A088613 A351582 A049723 Adjacent sequences: A035113 A035114 A035115 * A035117 A035118 A035119 KEYWORD nonn,easy,mult AUTHOR N. J. A. Sloane 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.)