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A035116
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a(n) = tau(n)^2, where tau(n) = A000005(n).
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30
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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)
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OFFSET
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1,2
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 304.
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LINKS
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FORMULA
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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).
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
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MAPLE
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A035116 := proc(n) numtheory[tau](n)^2 ; end proc:
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MATHEMATICA
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PROG
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(Magma) [ NumberOfDivisors(n)^2 : n in [1..100] ];
(Haskell)
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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