A348301 a(n) is the difference between the numerator and denominator of the (reduced) fraction Sum_{i = 1..n} 1/prime(i).

-1, -1, 1, 37, 617, 10331, 205657, 4417993, 111313529, 3451185211, 113456434771, 4398448576657, 187757129777747, 8377806843970331, 406839682998275587, 22177392981497097521, 1341055344385518798469, 83727136357670859345679, 5727006517323354547143763

Offset: 1

Author

Greg Tener, Oct 10 2021

Keywords

• sign

Examples

			a(1) = (p_1# / p_1) - p_1 = (2 / 2) - 2 = -1.
a(2) = (p_2# / p_1 + p_2# * p_2) - p_1 * p_2 = (6 / 2 + 6 / 3) - 2 * 3 = -1.
a(3) = 2*3*5/2 + 2*3*5/3 + 2*3*5/5 - 2*3*5 = 31 - 30 = 1.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A024451 (numerators), A002110 (denominators).

Programs

Mathematica

Numerator[#]-Denominator[#]&/@Accumulate[1/Prime[Range[20]]] (* Harvey P. Dale, Feb 05 2023 *)

PARI

a(n) = my(q=sum(i=1, n, 1/prime(i))); numerator(q)-denominator(q); \\ Michel Marcus, Oct 18 2021

Python

from itertools import islice
from sympy import primorial, sieve
def a(n): return sum(primorial(n) // p for p in islice(sieve, n)) - primorial(n) # Greg Tener, Oct 18 2021

Formula

a(n) = (Sum_{i = 1..n} p_n# / p_i) - p_n# where p_n# is the primorial of the n-th prime.

a(n) = A024451(n) - A002110(n).