A208133 -id:A208133 - cp's OEIS frontend

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

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

Offset: 1

Vaclav Kotesovec, Dec 19 2019

			1.106960111953217676651179130007439592949548833658122419043134044978777...

Cf. A065464, A065473, A065476, A208133, A328017, A330595, A330596.

Do[Print[N[Exp[-Sum[q = Expand[(-p^2 + 2*p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 + 1/p^2 - 2/p^3) \\ Amiram Eldar, Mar 16 2021

A335705 Decimal expansion of Sum_{primes p} 2(p-3) log(p) / (p^3 + p - 2).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 18 2020

			0.079339315977982136748150057589344450315501605856610561211269...

Cf. A208133, A306072, A335706.

ratfun = 2*(p-3) / (p^3 + p - 2); 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] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}]

A335706 Decimal expansion of Sum_{primes p} 2p(2p^3 - 9p^2 - 1) * log(p)^2 / (p^3 + p - 2)^2.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 18 2020

			0.23510971407787662832341660852337712786303845218859602743433...

Cf. A208133, A335705.

ratfun = 2*p*(2*p^3 - 9*p^2 - 1) / (p^3 + p - 2)^2; zetas = 0; ratab = Table[konfun = Together[Simplify[ratfun - c*(p^power/(p^power - 1)^2)]]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(-Zeta'[power]^2/Zeta[power]^2 + Zeta''[power]/Zeta[power]) /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]^2*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}]

A300828 Multiplicative with a(p^2) = 1, a(p^3) = -2 and a(p^e) = 0 when e = 1 or e > 3.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2

Offset: 1

Antti Karttunen, Mar 21 2018

Used for computing A208133.

			For n = 216 = 2^3 * 3^3, a(2^3) * a(3^3) = -2 * -2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A208133.

f[p_, e_] := Switch[e, 2, 1, 3, -2, , 0]; a[n] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

A300828(n) = { if(1==n, return(1)); my(val=1, v=factor(n), d=matsize(v)[1]); for(i=1,d, if(v[i,2] < 2 || v[i,2] > 3, return(0)); if (v[i,2] == 3, val *= -2)); return(val); }; \\ R. J. Mathar, Antti Karttunen, Mar 21 2018

;; With memoization-macro definec.
(definec (A300828 n) (if (= 1 n) n (* (case (A067029 n) ((2) 1) ((3) -2) (else 0)) (A300828 (A028234 n)))))

cp's OEIS Frontend

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A335705 Decimal expansion of Sum_{primes p} 2(p-3) log(p) / (p^3 + p - 2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A335706 Decimal expansion of Sum_{primes p} 2p(2p^3 - 9p^2 - 1) * log(p)^2 / (p^3 + p - 2)^2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A300828 Multiplicative with a(p^2) = 1, a(p^3) = -2 and a(p^e) = 0 when e = 1 or e > 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

cp's OEIS Frontend

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A335705 Decimal expansion of Sum_{primes p} 2*(p-3) * log(p) / (p^3 + p - 2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A335706 Decimal expansion of Sum_{primes p} 2*p*(2*p^3 - 9*p^2 - 1) * log(p)^2 / (p^3 + p - 2)^2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A300828 Multiplicative with a(p^2) = 1, a(p^3) = -2 and a(p^e) = 0 when e = 1 or e > 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A335705 Decimal expansion of Sum_{primes p} 2(p-3) log(p) / (p^3 + p - 2).

A335706 Decimal expansion of Sum_{primes p} 2p(2p^3 - 9p^2 - 1) * log(p)^2 / (p^3 + p - 2)^2.