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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. 21
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 (list; graph; refs; listen; history; text; internal format)
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
James Maynard, Small gaps between primes, arXiv:1311.4600 [math.NT], 2013-2019.
Eric Weisstein's World of Mathematics, de Polignac's Conjecture.
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
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
Sequence in context: A144309 A080376 A005669 * A261367 A148077 A148078
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

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)