A247951 -id:A247951 - cp's OEIS frontend

A345160 a(n) = Product_{k=1..n} sigma_3(k).

1, 9, 252, 18396, 2317896, 584109792, 200933768448, 117546254542080, 88982514688354560, 100906171656594071040, 134407020646583302625280, 274727950201616270566072320, 603852034543152562704226959360, 1869525898945600334132286666178560

Offset: 1

Vaclav Kotesovec, Jun 10 2021

nonn

Partial products of A001158.

Vaclav Kotesovec, Table of n, a(n) for n = 1..180
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

Cf. A001158, A066843, A066780, A247951, A345159.

Table[Product[DivisorSigma[3, k], {k, 1, n}], {n, 1, 20}]
FoldList[Times,DivisorSigma[3,Range[20]]] (* Harvey P. Dale, Sep 18 2023 *)

a(n) = prod(k=1, n, sigma(k, 3)); \\ Michel Marcus, Jun 10 2021

Lim_{n->infinity} (a(n) / (n!)^3)^(1/n) = A345159.

A345158 Product_{p primes, k>=1} ((p^(2k + 2) - 1)/(p^(2k + 2) - p^2))^(1/p^k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 10 2021

			1.19060271642680719164374162595970365073938314790455329116674114399345...

Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20), constant c.

Cf. A001157, A247951, A345144, A345159.

$MaxExtraPrecision = 1000; m = 600; prod = 1; s = 2; Do[Clear[f]; f[p_] := ((p^((k + 1)*s) - 1)/(p^((k + 1)*s) - p^s))^(1/p^k); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; prod *= f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]; Print[prod], {k, 1, 150}]

Equals lim_{n->infinity} (A247951(n) / (n!)^2)^(1/n).

cp's OEIS Frontend

A345160 a(n) = Product_{k=1..n} sigma_3(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A345158 Product_{p primes, k>=1} ((p^(2k + 2) - 1)/(p^(2k + 2) - p^2))^(1/p^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A345160 a(n) = Product_{k=1..n} sigma_3(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A345158 Product_{p primes, k>=1} ((p^(2*k + 2) - 1)/(p^(2*k + 2) - p^2))^(1/p^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A345158 Product_{p primes, k>=1} ((p^(2k + 2) - 1)/(p^(2k + 2) - p^2))^(1/p^k).