A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

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

Offset

1,1

Comments

m(k) can be proved to approach a harmonic series (and diverge) as k approaches infinity.

Links

Table of n, a(n) for n=1..104.

Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Formula

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (k^(j/k)*Gamma(j/k, 1/k))) where Gamma(a,x) the incomplete Gamma function.

Example

4.165243765558384590787262...
For n=10, the series is equal to 1+summation from n=1 to 10 (1/n)=9901/2520.

Mathematica

Multifactorial[n_, k_] := Abs[Apply[Times, Range[-n, -1, k]]]
N[Sum[1/Multifactorial[n, 10], {n, 0, 10000}], 105]
(* or *)
ReciprocalFactorialSumConstant[k_] :=
1/k Exp[1/k] (k + Sum[k^(j/k) Gamma[j/k, 0, 1/k], {j, k - 1}])
N[ReciprocalFactorialSumConstant[10], 105]

Crossrefs

Cf. A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Sequence in context: A228451 A324056 A120422 * A110312 A011242 A371498

Adjacent sequences: A342030 A342031 A342032 * A342034 A342035 A342036

Keyword

nonn,cons

Author

Bhoris Dhanjal, Feb 26 2021

Status

approved