A351219 -id:A351219 - cp's OEIS frontend

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

This is 1/Artin's constant, see A005596.

			2.67411272557002150896041183...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A048296, A351219.

$MaxExtraPrecision = 1200; digits = 99; terms = 1200; P[n_] := PrimeZetaP[n ]; LR = Join[{0, 0}, LinearRecurrence[{2, 0, -1}, {2, 3, 6}, term+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p^2-p-1)) \\ Amiram Eldar, Mar 15 2021

A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2p^(-2s)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 11, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 21, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 11, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 43, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 11, 11, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 21, 1, 3, 3, 9

Offset: 1

Ilya Gutkovskiy, Feb 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)

Cf. A001045, A061142, A351219, A351346, A351348.

f[p_, e_] := (2^(e + 1) + (-1)^e)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - 2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Multiplicative with a(p^e) = Jacobsthal(e+1).
From Vaclav Kotesovec, Feb 11 2022: (Start)
Let f(s) = Product_{prime p>2} (1 - 3/p^(2*s) + 2/p^(3*s))/(1 - 4/p^(2*s)), then
Sum_{k=1..n} a(k) ~ n*((2 * Pi^2 * log(n) + Pi^2 * (5*log(2) + 2*gamma - 2) + 24*zeta'(2))*f(1) + 2*Pi^2 * f'(1)) / (48*log(2)), where
f(1) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = A167864 = 1.5147801281374912577909192556494748924152701582862143953574842714849322098...,
f'(1) = -f(1) * Sum_{primes p > 2} 2*log(p) / (2 - 3*p + p^2) = -2*f(1)*A347195 = -2.603580548675394425068281893203286277011306183054394825715911358402698051... and gamma is the Euler-Mascheroni constant A001620. (End)

A351346 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2p^(-s) - p^(-2s)).

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 12, 5, 4, 2, 10, 2, 4, 4, 29, 2, 10, 2, 10, 4, 4, 2, 24, 5, 4, 12, 10, 2, 8, 2, 70, 4, 4, 4, 25, 2, 4, 4, 24, 2, 8, 2, 10, 10, 4, 2, 58, 5, 10, 4, 10, 2, 24, 4, 24, 4, 4, 2, 20, 2, 4, 10, 169, 4, 8, 2, 10, 4, 8, 2, 60, 2, 4, 10, 10, 4, 8, 2, 58, 29, 4, 2, 20, 4, 4, 4, 24, 2, 20

Offset: 1

Ilya Gutkovskiy, Feb 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)

Cf. A000129, A351219, A351347, A351348.

f[p_, e_] := Fibonacci[e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 90}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Multiplicative with a(p^e) = Pell(e+1).
From Vaclav Kotesovec, Feb 11 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^s, where
s = log(1 + sqrt(2)) / log(2) = 1.271553303163611972...,
c = 8.3717222015175571... = (1 + sqrt(2)) / (2^(3/2) * log(1 + sqrt(2))) * Product_{p primes > 2} 1 / (1 - 2*p^(-s) - p^(-2*s)),
or with better convergence
c = zeta(s)^2 / (sqrt(2) * (1 + sqrt(2)) * log(1 + sqrt(2))) * Product_{p primes > 2} (1 - p^(-s))^2 / (1 - 2*p^(-s) - p^(-2*s)). (End)

A351348 Dirichlet g.f.: Product_{p prime} (1 + 2p^(-s)) / (1 - p^(-s) - p^(-2s)).

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 7, 4, 9, 3, 12, 3, 9, 9, 11, 3, 12, 3, 12, 9, 9, 3, 21, 4, 9, 7, 12, 3, 27, 3, 18, 9, 9, 9, 16, 3, 9, 9, 21, 3, 27, 3, 12, 12, 9, 3, 33, 4, 12, 9, 12, 3, 21, 9, 21, 9, 9, 3, 36, 3, 9, 12, 29, 9, 27, 3, 12, 9, 27, 3, 28, 3, 9, 12, 12, 9, 27, 3, 33, 11, 9, 3, 36, 9, 9, 9

Offset: 1

Ilya Gutkovskiy, Feb 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Cf. A000204, A074823, A351219, A351346, A351347.

f[p_, e_] := LucasL[e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 87}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Multiplicative with a(p^e) = Lucas(e+1).
a(n) = Sum_{d|n} A074823(d) * A351219(n/d).
From Vaclav Kotesovec, Feb 12 2022: (Start)
Let f(s) = Product_{p prime} (1 + 1/(p^(2*s) - p^s - 1)) * (1 - 3/p^(2*s) + 2/p^(3*s)), then
Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + ((3*g-1)*f(1) + f'(1))*log(n) + (1 - 3*g + 3*g^2 - 3*sg1)*f(1) + (3*g-1)*f'(1) + f''(1)/2), where
f(1) = Product_{prime p} (p-1)^3 * (p+2) / (p^2 (p^2 - p - 1)) = 0.76679494740111861346654669603448358442373234633770198438779408968851774...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 - 9*p - 4) * log(p) / (p^4 - 4*p^2 + p + 2) = -0.2518173642312369311596467494348076414732211832249275289370643712012051...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{p prime} -p*(8*p^5 - 27*p^4 - 16*p^3 + 32*p^2 + 16*p + 14) * log(p)^2 / (p^4 - 4*p^2 + p + 2)^2 = 4.28643633804365513728313780779157573071314496047204449783182235740130206...,
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A351655 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - p^(-2s) - p^(-3s)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 7, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 13, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 24, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 7, 7, 1, 1, 2, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, Feb 16 2022

Cf. A000073, A351219, A351346, A351347, A351348, A351656.

t[n_] := t[n] = t[n-1] + t[n-2] + t[n-3]; t[0] = t[1] = 0; t[2] = 1; f[p_, e_] := t[e+2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

for(n=1, 87, print1(direuler(p=2, n, 1/(1 - X - X^2 - X^3))[n], ", "))

Multiplicative with a(p^e) = A000073(e+2).

A351656 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - p^(-2s) - p^(-3s) - p^(-4*s)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 15, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 29, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, Feb 16 2022

Cf. A000078, A351219, A351346, A351347, A351348, A351655.

t[e_] := t[e] = If[e < 5, 2^(e-1), t[e-1] + t[e-2] + t[e-3] + t[e-4]]; a[1] = 1; a[n_] := Times @@ t /@ Last @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

for(n=1, 87, print1(direuler(p=2, n, 1/(1 - X - X^2 - X^3 - X^4))[n], ", "))

Multiplicative with a(p^e) = A000078(e+3).

cp's OEIS Frontend

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

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A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2*p^(-2*s)).

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A351346 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2*p^(-s) - p^(-2*s)).

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A351348 Dirichlet g.f.: Product_{p prime} (1 + 2*p^(-s)) / (1 - p^(-s) - p^(-2*s)).

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A351655 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - p^(-2*s) - p^(-3*s)).

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A351656 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - p^(-2*s) - p^(-3*s) - p^(-4*s)).

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A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2p^(-2s)).

A351346 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2p^(-s) - p^(-2s)).

A351348 Dirichlet g.f.: Product_{p prime} (1 + 2p^(-s)) / (1 - p^(-s) - p^(-2s)).

A351655 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - p^(-2s) - p^(-3s)).

A351656 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - p^(-2s) - p^(-3s) - p^(-4*s)).