A335818 Decimal expansion of Sum_{k>=1} 1/phi(k)^3, where phi is the Euler totient function.

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

Offset: 1

Author

Vaclav Kotesovec, Jun 25 2020

Keywords

• nonn
• cons

Comments

Sum_{k>=1} 1/phi(k)^m is convergent iff m > 1 (reference Monier). - Bernard Schott, Jan 14 2021

Examples

			2.476194748165025794326855444125145160045456856355284384345707879150949...
		

References

• Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294.

Links

• Eric Weisstein's World of Mathematics, Totient Function.
• Eric Weisstein's World of Mathematics, Totient Summatory Function.
• Wikipedia, Euler's totient function.

Crossrefs

Cf. A000010, A065484, A109695, A335319.

Programs

Mathematica

$MaxExtraPrecision = 1000; f[p_] := (1 + 1/((1 - 1/p)^s * (p^s - 1))) /. s -> 3; Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {m, 100, 1000, 100}]

PARI

prodeulerrat(1 + 1/((1 - 1/p)^3 * (p^3 - 1))) \\ Amiram Eldar, Mar 15 2021

Formula

Equals Product_{primes p} (1 + 1/((1 - 1/p)^3 * (p^3 - 1))).