A332713 -id:A332713 - cp's OEIS frontend

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

5, 3, 5, 8, 9, 6, 1, 5, 3, 8, 2, 8, 3, 3, 7, 9, 9, 9, 8, 0, 8, 5, 0, 2, 6, 3, 1, 3, 1, 8, 5, 4, 5, 9, 5, 0, 6, 4, 8, 2, 2, 2, 3, 7, 4, 5, 1, 4, 1, 4, 5, 2, 7, 1, 1, 5, 1, 0, 1, 0, 8, 3, 4, 6, 1, 3, 3, 2, 8, 8, 1, 1, 9, 6, 1, 4, 5, 4, 1, 1, 0, 4, 5, 1, 1, 4, 4, 6, 5, 8, 2, 7, 3, 1, 0, 0, 2, 3, 4, 4, 0, 5, 3, 5, 1, 1

Offset: 0

Vaclav Kotesovec, Dec 17 2019

			0.5358961538283379998085026313185459506482223745141452711510108346133288119...

Cf. A055653, A062354, A065465, A175639, A256392, A332713.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 + p^3 - p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}]

(6/Pi^2) * prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Sep 08 2020

Equals (6/Pi^2) * A065465. - Amiram Eldar, Sep 08 2020

A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A001694 (positions of nonzero terms), A005361, A007427, A008683, A008836, A028242, A052485 (positions of 0's), A062838 (positions of 1's), A112526, A252505, A322483, A332685, A332713.

Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]
A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
f[p_, e_] := 3*Floor[e/2] - e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(2*s)^2 * zeta(3*s) / zeta(6*s).
a(n) = Sum_{d|n} mu(lcm(d, n/d)/d).
a(n) = Sum_{d|n} (-1)^bigomega(n/d) * A005361(d).
a(n) = Sum_{d|n} A010052(n/d) * A112526(d).
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)*log(n)/(2*zeta(3)) + ((2*gamma - 1)*zeta(3/2) + 3*zeta'(3/2)/2 - 3*zeta(3/2)*zeta'(3)/zeta(3)) * sqrt(n)/zeta(3) + 6*zeta(2/3)^2 * n^(1/3)/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 21 2020
Multiplicative with a(p^e) = A028242(e). - Amiram Eldar, Nov 30 2020

A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 8, 5, 9, 3, 15, 3, 9, 9, 11, 3, 15, 3, 15, 9, 9, 3, 24, 5, 9, 8, 15, 3, 27, 3, 15, 9, 9, 9, 25, 3, 9, 9, 24, 3, 27, 3, 15, 15, 9, 3, 33, 5, 15, 9, 15, 3, 24, 9, 24, 9, 9, 3, 45, 3, 9, 15, 19, 9, 27, 3, 15, 9, 27, 3, 40, 3, 9, 15

Offset: 1

Ilya Gutkovskiy, Feb 21 2020

Inverse Moebius transform of A322483.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A007425, A024206, A048691, A124315, A124316, A295316, A322483, A332619, A332713.

Table[Sum[DivisorSigma[0, d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e+3)/2]; A322483[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Table[Sum[A322483[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e + 1)*(e + 5)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n) = Sum_{d|n} A322483(d).
a(n) = Sum_{d|n} tau(n/d) * A295316(d).
Multiplicative with a(p^e) = floor((e+1)*(e+5)/4) = A024206(e+2). - Amiram Eldar, Dec 05 2022

cp's OEIS Frontend

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

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A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

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A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

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