A065484 Decimal expansion of Product_{p prime >= 2} (1 + p/((p-1)^2*(p+1))).

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

Offset: 1

Author

N. J. A. Sloane, Nov 19 2001, Aug 09 2010

Keywords

• cons
• nonn

Comments

Decimal expansion of totient constant. - Eric W. Weisstein, Apr 20 2006

Examples

			2.203856596437859787872828316480...
		

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• Eric Weisstein's World of Mathematics, Totient Summatory Function.

Crossrefs

Cf. A000010, A002618, A065483, A077387.

Programs

Mathematica

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n];
LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]];  (Pi^2/6)*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 *)

PARI

prodeulerrat(1 + p/((p-1)^2*(p+1))) \\ Hugo Pfoertner, Jun 02 2020

Formula

Equals Pi^2 * A065483 / 6.

Also defined as: Sum_{m>=1} 1/(m*A000010(m)). See Weisstein link.

Equals 5 * Sum_{m>=1} (-1)^(m+1)/(m*A000010(m)). - Amiram Eldar, Nov 21 2022