A038664 a(n)-th and (a(n)+1)-st primes are the first pair of primes that differ by exactly 2n; a(n) = -1 if no such pair of primes exists.

2, 4, 9, 24, 34, 46, 30, 282, 99, 154, 189, 263, 367, 429, 590, 738, 217, 1183, 3302, 2191, 1879, 1831, 7970, 3077, 3427, 2225, 3793, 8028, 4612, 4522, 3644, 8688, 14862, 12542, 15783, 3385, 34202, 19026, 17006, 44773, 23283, 38590, 14357

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Does anyone know of a proof that a(n) is defined for all natural numbers n, i.e., f:n -> prime(n+1)-prime(n) is a surjective map from N-{1} -> E, where N, E are the sets of natural numbers and even numbers, respectively? - Joseph L. Pe, Dec 14 2002

a(n) is defined for all n if (but not only if) de Polignac's conjecture is true. - Harry J. Smith, Jul 22 2003

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..111
• James Maynard, Small gaps between primes, arXiv:1311.4600 [math.NT], 2013-2019.
• Eric Weisstein's World of Mathematics, de Polignac's Conjecture.

Crossrefs

Cf. A000230, A000720, A001223, A261525.

Programs

Haskell

import Data.List (elemIndex); import Data.Maybe (fromJust)
a038664 = (+ 1) . fromJust . (`elemIndex` a001223_list) . (* 2)
-- Reinhard Zumkeller, Aug 23 2015

Mathematica

Table[k = 0; While[k++; p1 = Prime[k]; p2 = Prime[k + 1]; (p2 - p1) != n]; k, {n, 2, 200, 2}] (* Lei Zhou, Mar 01 2005 *)
With[{d=Differences[Prime[Range[50000]]]},Flatten[Table[Position[d,2n,1,1],{n,50}]]] (* This program is many times faster than the first Mathematica program above. *) (* Harvey P. Dale, Nov 24 2012 *)

PARI

first(m)=my(v=vector(m),n);for(n=1,m,v[n]=0;until(2*n==prime(v[n]+1)-prime(v[n]),v[n]++)); v; \\ Anders Hellström, Jul 19 2015

Formula

a(n) = A000720(A000230(n)). - M. F. Hasler, Jan 16 2011

A001223(a(n)) = 2*n and A001223(m) != 2*n for m < a(n). - Reinhard Zumkeller, Aug 23 2015

Extensions

More terms from Michel ten Voorde, Apr 13 2001

"a(n) = -1 if ..." added to definition at the suggestion of Alexander Wajnberg by N. J. A. Sloane, Feb 02 2020