A351347 -id:A351347 - cp's OEIS frontend

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

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)

A347195 Decimal expansion of Sum_{primes p > 2} log(p) / ((p-2)*(p-1)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 22 2021

Constant is related to the asymptotics of A069205.

			0.8593922313585686897187145141861232817699609176983112114741634265903839649...

Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44. [constant C3]

Cf. A069205, A351347.

ratfun = 1/((p-2)*(p-1)); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(Zeta'[power]/Zeta[power] + Log[2]/(2^power - 1)) /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 2, m}] + zetas, 110]], {m, 2000, 10000, 2000}]

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

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A347195 Decimal expansion of Sum_{primes p > 2} log(p) / ((p-2)*(p-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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)).