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A347961
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Dirichlet convolution of A342001 with itself.
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5
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0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 14, 0, 2, 2, 10, 0, 14, 0, 18, 2, 2, 0, 42, 1, 2, 4, 22, 0, 40, 0, 20, 2, 2, 2, 63, 0, 2, 2, 58, 0, 48, 0, 30, 20, 2, 0, 92, 1, 18, 2, 34, 0, 40, 2, 74, 2, 2, 0, 204, 0, 2, 24, 35, 2, 64, 0, 42, 2, 56, 0, 162, 0, 2, 20, 46, 2, 72, 0, 132, 10, 2, 0, 260, 2, 2, 2, 106, 0, 210, 2, 54, 2, 2, 2
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OFFSET
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1,6
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LINKS
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FORMULA
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Let pr(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))
and su(s) = Sum_{primes p} p^s/((p^s - 1)*(p^s + p - 1)).
Sum_{k=1..n} a(k) ~ pr(2)^2 * su(2)^2 * Pi^4 * n^2 * log(n) / 72 *
(1 + (2*gamma - 1/2 + 2*pr'(2)/pr(2) + 2*su'(2)/su(2) + 12*zeta'(2)/Pi^2) / log(n)), where
pr(2) = A065464 = 0.428249505677094440218765707581823546121298513355936...
pr'(2) = pr(2) * Sum_{primes p} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407...
su(2) = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2)) * PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384...
su'(2) = Sum_{primes p} p^2 * (1-p-p^4) * log(p) / ((p^2-1)^2 * (p^2+p-1)^2)) = -0.486606220169261905698805096547122238460686354267440350206456696497...
and gamma is the Euler-Mascheroni constant A001620. (End)
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PROG
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(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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