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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. 16
 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 A001223(a(n)) = 2*n and A001223(m) != 2*n for m < a(n). - Reinhard Zumkeller, Aug 23 2015 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..111 Eric Weisstein's World of Mathematics, de Polignac's Conjecture. FORMULA a(n) = A000720(A000230(n)). - M. F. Hasler, Jan 16 2011 MAPLE P:=proc(q) local k, n; for k from 2 by 2 to q do for n from 1 to q do if ithprime(n+1) mod ithprime(n)=k then print(n); break; fi; od; od; end: P(10^9); # Paolo P. Lava, Mar 22 2017 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]]}, 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 Cf. A001223, A261525. Sequence in context: A144309 A080376 A005669 * A261367 A148077 A148078 Adjacent sequences:  A038661 A038662 A038663 * A038665 A038666 A038667 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 April 8 12:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)