A083281 Decimal expansion of h = Product_{p prime}(sqrt(p(p-1))*log(1/(1-1/p))).

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

Offset

0,1

Comments

Arises in formulas like: Sum_{k<=x} 1/tau(kd) = hx/sqrt(Pi*log(x))*{ g(d)+O((3/4)^omega(d)/log(x)) } where g satisfies Sum_{d<=x} g(d))=x/h/sqrt(Pi*log(x))*{ 1+O(1/log(x)) }.

The logarithm of the value has an expansion -P(2)/24 -P(3)/24 -109*P(4)/2880 -49*P(5)/1440-... in terms of the prime zeta functions P(.). - R. J. Mathar, Jan 31 2009

The average order of 1/tau(k) (where tau(k) is the number of divisors of k, A000005), Sum_{k<=x} 1/tau(k) ~ h*x/sqrt(Pi*log(x)), was found by Ramanujan in 1916 and was proven by Wilson in 1923. - Amiram Eldar, Jun 19 2019

References

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, collection SMF no. 1, 1995, p. 210.

Links

Table of n, a(n) for n=0..105.

S. Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

B. M. Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 235-255.

Formula

Equals A345231 * sqrt(Pi). - Vaclav Kotesovec, Jun 13 2021

Example

0.96927694382749163071695371472090732266213688638491621816178588751950570028...

Mathematica

$MaxExtraPrecision = 1000; m = 1000; f[p_] := Sqrt[p*(p - 1)]*Log[p/(p - 1)]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Jun 19 2019 *)

Prog

(PARI) prod(k=1, 40000, sqrt(prime(k)*(prime(k)-1))*log(1/(1-1/prime(k))))

Crossrefs

Cf. A000005, A345231, A345288.

Sequence in context: A363539 A086279 A155533 * A019711 A344778 A010545

Adjacent sequences: A083278 A083279 A083280 * A083282 A083283 A083284

Keyword

cons,nonn

Author

Benoit Cloitre, Jun 02 2003

Extensions

10 more digits from R. J. Mathar, Jan 31 2009

More terms from Amiram Eldar, Jun 19 2019

More digits from Vaclav Kotesovec, Jun 13 2021

Status

approved