A064647 -id:A064647 - cp's OEIS frontend

A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

1, 3, 15, 105, 189, 10395, 135135, 2027025, 34459425, 130945815, 13749310575, 316234143225, 7905853580625, 12556355686875, 1238056670725875, 776918153694375, 6332659870762850625, 7642865361265509375, 8200794532637891559375, 63966197354575554163125, 13113070457687988603440625

Offset: 1

Ilya Gutkovskiy, May 23 2022

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A001147, A053556, A061355, A064647, A113013, A143383, A289488, A306858, A354298 (numerators).

S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, denom(S);
od:
R; # Robert Israel, Jan 10 2024

Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Denominator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Denominator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Denominator

Denominators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

A064646 Numerators of partial sums of reciprocals of primorial numbers.

Original entry on oeis.org

1, 2, 7, 74, 543, 10589, 120009, 3420257, 4767631, 2281311434, 141441308909, 51307141467, 3301022547923, 200573000466191, 433638827007904943, 328326540448842314, 31534618884970203647, 13785884222546140694347

Offset: 1

Labos Elemer, Oct 04 2001

			For n = 5, Sum_{j=1..5} 1/A002110(j) = 1/2 + 1/6 + 1/30 + 1/210 + 1/2310 = (1155 + 385 + 77 + 11 + 1)/2310 = 1629/2310 = 543/770, so a(5) = 543.

Cf. A002110, A064647 (denominators), A064648.

q[x_] := Apply[Times, Table[Prime[j], {j, 1, x}]]; a[n_] := Numerator[Apply[Plus, Table[1/q[w], {w, 1, n}]]]; Array[a, 18]

list(lim) = {my(s = 0, pr = 1); forprime(p = 1, lim, pr *= p; s += 1/pr; print1(numerator(s), ", "));} \\ Amiram Eldar, Feb 08 2025

Limit_{n->oo} a(n)/A064647(n) = A064648. - Amiram Eldar, Feb 08 2025

cp's OEIS Frontend

A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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