OFFSET
1,1
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.
FORMULA
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 *)
PROG
(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
(Haskell)
import Data.List (elemIndex); import Data.Maybe (fromJust)
a038664 = (+ 1) . fromJust . (`elemIndex` a001223_list) . (* 2)
-- Reinhard Zumkeller, Aug 23 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
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
STATUS
approved