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A038664
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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.
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21
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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"a(n) = -1 if ..." added to definition at the suggestion of Alexander Wajnberg by N. J. A. Sloane, Feb 02 2020
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STATUS
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approved
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