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A158949
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Inverse Moebius transform of A065958.
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1
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1, 6, 11, 26, 27, 66, 51, 106, 101, 162, 123, 286, 171, 306, 297, 426, 291, 606, 363, 702, 561, 738, 531, 1166, 677, 1026, 911, 1326, 843, 1782, 963, 1706, 1353, 1746, 1377, 2626, 1371, 2178, 1881, 2862, 1683, 3366, 1851, 3198, 2727, 3186, 2211, 4686, 2501
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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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a(n) = (1/n^2)*Sum_{d|n} sigma_2(d)^2*moebius(n/d).
a(n) = Sum_{d|n} 2^omega(n/d) * d^2. - Daniel Suteu, Mar 07 2019
Multiplicative with a(p^e) = (p^(2*e)*(p^2+1) - 2)/(p^2-1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)^2/(3*zeta(6)) = 0.473436... . (End)
Dirichlet g.f.: zeta(s)^2*zeta(s-2)/zeta(2*s). - Amiram Eldar, Jan 06 2023
a(n) = Sum_{1 <= j, k <= n} tau(gcd(j, k, n)^2) = Sum_{d divides n} tau(d^2)* J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024
a(n) = Sum_{d divides n} J_4(d)/J_2(d) = Sum_{1 <= i, j, k, l <= n} 1/(J_2(n/gcd(i,j,k,l,n))), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024
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MAPLE
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A158949 := proc(n) add(numtheory[sigma][2](d)^2*numtheory[mobius](n/d), d=numtheory[divisors](n))/n^2 ; end: seq( A158949(n), n=1..80) ; # R. J. Mathar, Apr 02 2009
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MATHEMATICA
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a[n_] := Sum[2^PrimeNu[n/d] d^2, {d, Divisors[n]}];
f[p_, e_] := (p^(2*e)*(p^2 + 1) - 2)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Dec 05 2022 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, 2^omega(n/d) * d^2); \\ Daniel Suteu, Mar 07 2019
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CROSSREFS
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KEYWORD
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easy,mult,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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