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A034714
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Dirichlet convolution of squares with themselves.
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8
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1, 8, 18, 48, 50, 144, 98, 256, 243, 400, 242, 864, 338, 784, 900, 1280, 578, 1944, 722, 2400, 1764, 1936, 1058, 4608, 1875, 2704, 2916, 4704, 1682, 7200, 1922, 6144, 4356, 4624, 4900, 11664, 2738, 5776, 6084, 12800, 3362, 14112, 3698, 11616, 12150, 8464
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OFFSET
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1,2
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LINKS
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FORMULA
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Dirichlet g.f.: zeta^2(s-2).
Equals n^2*tau(n), where tau(n) = A000005(n) = number of divisors of n. - Jon Perry, Aug 28 2005
Multiplicative with a(p^e) = (e+1)p^(2e). - Mitch Harris, Jun 27 2005
G.f.: Sum_{k>=1} k^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (-p^2 * log(1 - 1/p^2)) = 1.27728092754165872535305748273941301416624226497497308879403022758421224... - Vaclav Kotesovec, Sep 19 2020
G.f.: Sum_{n >= 1} q^(n^2)*( n^4*q^(3*n) - n^2*(n^2 + 4*n - 2)*q^(2*n) - n^2*(n^2 - 4*n - 2)*q^n + n^4 )/(1 - q^n)^3 - apply the operator q*d/dq twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
Sum_{k=1..n} a(k) ~ (n^3/3) * (log(n) + 2*gamma - 1/3), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 02 2023
a(n) = Sum_{1 <= i, j <= n} sigma_2( gcd(i, j, n) ) = Sum_{d divides n} sigma_2(d) * J_2(n/d), where sigma_2(n) = A001157(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024
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MAPLE
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A034714 := proc(n) n^2*numtheory[tau](n) ; end proc:
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MATHEMATICA
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PROG
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(Magma) [n^2*NumberOfDivisors(n): n in [1..50]]; // Bruno Berselli, Feb 12 2014
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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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STATUS
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approved
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