1,2

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

Contribution from Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 23 2008: (Start)

Shinya: Let p_{k} [A000040(k)] denote the k-th prime and

d(p_{k}) = p_{k} - p_{k - 1}, [A001223(k)] the difference between consecutive

primes. We denote by N_{epsilon}(x) the number of primes =< x which satisfy

the inequality d(p_{k}) =< )log p_{k})^(2 + epsilon), where epsilon > 0 is

arbitrary and fixed and by pi(x) [A000720(x)] the number of primes less than

or equal to x. In this paper, we prove that N(x)/pi(x) ~ 1 as x approaches infinity. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

K. Soundararajan, Small gaps bewteen prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.

N. J. A. Sloane, First 10000 terms

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

S. Ares & M. Castro, Hidden structure in the randomness of the prime number sequence ?

D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between primes and almost primes

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for primes, gaps between

Hisanobu Shinya, On the density of prime differences less than a given magnitude which satisfy a certain inequality, Sep 19, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 23 2008]

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

with(numtheory): for n from 1 to 500 do printf(`%d, `, ithprime(n+1) - ithprime(n)) od:

p = Table[Prime[i], {i, 1, 100}]; Drop[p, 1] - Drop[p, -1]

(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

Cf. A000040, A037201, A007921, A030173. Second difference is A036263, First occurrence is A000230.

Cf. A036263-A036274.

Sequence in context: A082508 A163824 A075526 this_sequence A118776 A092520 A147848

Adjacent sequences: A001220 A001221 A001222 this_sequence A001224 A001225 A001226

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001

Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2009