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Search: id:A001223
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| A001223 |
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Differences between consecutive primes.
(Formerly M0296 N0108)
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+0
276
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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
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REFERENCES
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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.
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LINKS
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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, December 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
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FORMULA
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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
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MAPLE
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with(numtheory): for n from 1 to 500 do printf(`%d, `, ithprime(n+1) - ithprime(n)) od:
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MATHEMATICA
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p = Table[Prime[i], {i, 1, 100}]; Drop[p, 1] - Drop[p, -1]
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PROGRAM
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(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
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CROSSREFS
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Cf. A000040, A037201, A007921, A030173. Second difference is A036263, First occurrence is A000230.
Cf. A036263-A036274.
Adjacent sequences: A001220 A001221 A001222 this_sequence A001224 A001225 A001226
Sequence in context: A106469 A082508 A075526 this_sequence A118776 A092520 A129089
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KEYWORD
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nonn,nice,easy
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AUTHOR
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njas
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001
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