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a(n) is the difference between the numerator and denominator of the (reduced) fraction Sum_{i = 1..n} 1/prime(i).
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%I #25 Feb 05 2023 17:56:46

%S -1,-1,1,37,617,10331,205657,4417993,111313529,3451185211,

%T 113456434771,4398448576657,187757129777747,8377806843970331,

%U 406839682998275587,22177392981497097521,1341055344385518798469,83727136357670859345679,5727006517323354547143763

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

%H Harvey P. Dale, <a href="/A348301/b348301.txt">Table of n, a(n) for n = 1..350</a>

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

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

%e a(1) = (p_1# / p_1) - p_1 = (2 / 2) - 2 = -1.

%e a(2) = (p_2# / p_1 + p_2# * p_2) - p_1 * p_2 = (6 / 2 + 6 / 3) - 2 * 3 = -1.

%e a(3) = 2*3*5/2 + 2*3*5/3 + 2*3*5/5 - 2*3*5 = 31 - 30 = 1.

%t Numerator[#]-Denominator[#]&/@Accumulate[1/Prime[Range[20]]] (* _Harvey P. Dale_, Feb 05 2023 *)

%o (Python)

%o from itertools import islice

%o from sympy import primorial, sieve

%o def a(n): return sum(primorial(n) // p for p in islice(sieve, n)) - primorial(n) # _Greg Tener_, Oct 18 2021

%o (PARI) a(n) = my(q=sum(i=1, n, 1/prime(i))); numerator(q)-denominator(q); \\ _Michel Marcus_, Oct 18 2021

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

%K sign

%O 1,4

%A _Greg Tener_, Oct 10 2021