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A082476
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a(n) = Sum_{d|n} mu(d)^2*tau(d)^2.
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6
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1, 5, 5, 5, 5, 25, 5, 5, 5, 25, 5, 25, 5, 25, 25, 5, 5, 25, 5, 25, 25, 25, 5, 25, 5, 25, 5, 25, 5, 125, 5, 5, 25, 25, 25, 25, 5, 25, 25, 25, 5, 125, 5, 25, 25, 25, 5, 25, 5, 25, 25, 25, 5, 25, 25, 25, 25, 25, 5, 125, 5, 25, 25, 5, 25, 125, 5, 25, 25, 125, 5, 25, 5, 25, 25, 25, 25, 125
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listen;
history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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More generally : sum(d|n, mu(d)^2*tau(d)^m) = (2^m+1)^omega(n).
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LINKS
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FORMULA
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a(n) = 5^omega(n); multiplicative with a(p^e)=5.
Dirichlet g.f.: Product_{primes p} (1 + 5/(p^s - 1)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 10/p^(2*s) + 20/p^(3*s) - 15/p^(4*s) + 4/p^(5*s)), (with a product that converges for s=1). (End)
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MATHEMATICA
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tau[1, n_] := 1; SetAttributes[tau, Listable];
tau[k_, n_] := Plus @@ (tau[k - 1, Divisors[n]]) /; k > 1;
(* or more easy *)
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PROG
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(PARI) a(n)=5^omega(n)
(PARI) for(n=1, 100, print1(direuler(p=2, n, (4*X+1)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 28 2023
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CROSSREFS
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KEYWORD
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mult,nonn
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AUTHOR
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STATUS
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approved
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