%I A001223 M0296 N0108
%S A001223 1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,
%T A001223 2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,6,4,2,10,14,4,2,4,
%U A001223 14,6,10,2,4,6,8,6,6,4,6,8,4,8,10,2,10,2,6,4,6,8,4,2,4,12,8,4,8,4,6,12
%N A001223 Differences between consecutive primes.
%C A001223 There is a unique decomposition of the primes: provided the weight A117078(n) 
               is > 0, we have prime(n) = weight * level + gap, or A000040(n) = 
               A117078(n) * A117563(n) + a(n). - Remi Eismann (reismann(AT)free.fr), 
               Feb 14 2008
%C A001223 Contribution from Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 23 
               2008: (Start)
%C A001223 Shinya: Let p_{k} [A000040(k)] denote the k-th prime and
%C A001223 d(p_{k}) = p_{k} - p_{k - 1}, [A001223(k)] the difference between consecutive
%C A001223 primes. We denote by N_{epsilon}(x) the number of primes =< x which satisfy
%C A001223 the inequality d(p_{k}) =< )log p_{k})^(2 + epsilon), where epsilon > 
               0 is
%C A001223 arbitrary and fixed and by pi(x) [A000720(x)] the number of primes less 
               than
%C A001223 or equal to x. In this paper, we prove that N(x)/pi(x) ~ 1 as x approaches 
               infinity. (End)
%D A001223 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001223 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001223 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 870.
%D A001223 K. Soundararajan, Small gaps bewteen prime numbers: the work of Goldston-Pintz-Yildirim, 
               Bull. Amer. Math. Soc., 44 (2007), 1-18.
%H A001223 N. J. A. Sloane, <a href="b001223.txt">First 10000 terms</a>
%H A001223 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001223 S. Ares & M. Castro, <a href="http://arXiv.org/abs/cond-mat/0310148">
               Hidden structure in the randomness of the prime number sequence ?</
               a>
%H A001223 D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, <a href="http:/
               /arXiv.org/abs/math.NT/0506067">Small gaps between primes and almost 
               primes</a>
%H A001223 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AndricasConjecture.html">Link to a section of The World of Mathematics.</
               a>
%H A001223 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeDifferenceFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A001223 <a href="Sindx_Pri.html#gaps">Index entries for primes, gaps between</
               a>
%H A001223 Hisanobu Shinya, <a href="http://arxiv.org/abs/0809.3458">On the density 
               of prime differences less than a given magnitude which satisfy a 
               certain inequality</a>, Sep 19, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Sep 23 2008]
%F A001223 G.f. b(x)*(1-x), where b(x) is the g.f. for the primes. - Frank Adams-Watters 
               (FrankTAW(AT)Netscape.net), Jun 15 2006
%p A001223 with(numtheory): for n from 1 to 500 do printf(`%d,`,ithprime(n+1) - 
               ithprime(n)) od:
%t A001223 p = Table[Prime[i], {i, 1, 100}]; Drop[p, 1] - Drop[p, -1]
%o A001223 (SAGE) v = primes_first_n(98) list = [] for i in range(97): list.append(v[1+i]-v[i]) 
               list - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2007
%Y A001223 Cf. A000040, A037201, A007921, A030173. Second difference is A036263, 
               First occurrence is A000230.
%Y A001223 Cf. A036263-A036274.
%Y A001223 Sequence in context: A082508 A163824 A075526 this_sequence A118776 A092520 
               A147848
%Y A001223 Adjacent sequences: A001220 A001221 A001222 this_sequence A001224 A001225 
               A001226
%K A001223 nonn,nice,easy
%O A001223 1,2
%A A001223 N. J. A. Sloane (njas(AT)research.att.com).
%E A001223 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001
%E A001223 Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 07 2009