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Denominator of the alternating n-th partial sum of the reciprocals of the successive prime gaps.
2

%I #15 Aug 02 2024 19:00:56

%S 1,2,1,4,4,1,2,4,12,12,12,6,3,12,4,12,12,12,3,6,3,12,4,8,8,8,8,8,8,56,

%T 56,168,168,840,840,840,840,840,280,840,840,840,840,840,840,280,840,

%U 840,840,840,280,280,280,840,280,840,840,840,840,840,168,168,168,168,168,168,168

%N Denominator of the alternating n-th partial sum of the reciprocals of the successive prime gaps.

%H Robert Israel, <a href="/A274980/b274980.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Denominator(Sum_{i=1..n} ((-1)^(i - 1))/(prime(i+1)-prime(i))).

%F a(n) = Denominator(Sum_{i=1..n} ((-1)^(i - 1))/A001223(i)).

%p P:= [seq(ithprime(i),i=1..101)]:

%p G:= P[2..-1]-P[1..-2]:

%p R:= ListTools:-PartialSums([seq((-1)^i/G[i],i=1..100)]):

%p map(denom,R); # _Robert Israel_, Aug 02 2024

%t Table[Denominator@Sum[((-1)^(j - 1))/(Prime[j + 1] - Prime[j]), {j, 1, n}], {n, 1, 120}];

%Y Cf. A001223, A274828, A275056.

%K nonn,frac,look

%O 1,2

%A _Andres Cicuttin_, Jul 14 2016