A286634 Numerator of the ratio of alternate consecutive prime gaps: Numerator ((prime(n + 3) - prime(n + 2))/(prime(n + 1) - prime(n))).

2, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 7, 1, 3, 1, 5, 1, 3, 3, 2, 1, 3, 1, 5, 1, 2, 1, 3, 6, 1, 1, 1, 3, 1, 5, 3, 3, 1, 1, 1, 2, 1, 5, 7, 2, 1, 1, 7, 3, 5, 1, 2, 3, 2, 1, 3, 2, 1, 2, 2, 1, 5, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 6, 2, 1, 1, 1, 3, 3, 1, 3, 3, 5, 1

Offset

1,1

Links

Table of n, a(n) for n=1..100.

Formula

a(n) = numerator((prime(n + 3) - prime(n + 2))/(prime(n + 1) - prime(n))).

A000040(n+3) = 5 + Sum_{k=1..n} ((1+(-1)^k)*Product_{j=1..k}(a(j)/A285851(j))^((1+(-1)^j)/2) + ((1-(-1)^k)/2)*Product_{j=1..k}(a(j)/A285851(j))^((1-(-1)^j)/2)), for n>0.

A001223(n+2) = (1+(-1)^n)*Product_{j=1..n}(a(j)/A285851(j))^((1+(-1)^j)/2) - ((-1+(-1)^n)/2)*Product_{j=1..n}(a(j)/A285851(j))^((1-(-1)^j)/2), for n>0

Mathematica

Table[Numerator[(Prime[k+3]-Prime[k+2])/(Prime[k+1]-Prime[k])], {k, 100}]

Crossrefs

Cf. A285851 (denominators), A001223, A000040, A272863, A276309.

Sequence in context: A060990 A276309 A165031 * A099245 A185331 A206474

Adjacent sequences: A286631 A286632 A286633 * A286635 A286636 A286637

Keyword

nonn,frac

Author

Andres Cicuttin, May 11 2017

Status

approved