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A001223
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Differences between consecutive primes.
(Formerly M0296 N0108)
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373
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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; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
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 <= x. In this paper, we prove that N(x)/pi(x) ~ 1 as x approaches infinity. [Jonathan Vos Post, Sep 23 2008]
Goldston et al prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes. [Jonathan Vos Post, Mar 21, 2011].
Goldston & Ledoan refine one aspect of a theorem of Gallagher that the prime k-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing. [Jonathan Vos Post, Nov 15, 2011]
Let rho(m) = A179196(m), for any n, let m be an integer such that p_(rho(m)) <= p_n and p_(n+1) <= p_(rho(m+1)), then rho(m) <= n < n + 1 <= rho(m + 1), therefore a(n) = p_(n+1) - p_n <= p_rho(m+1) - p_rho(m) = A182873(m). For all rho(m) = A179196(m), a(rho(m)) < A165959(m). [John W. Nicholson, Dec 14 2011]
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REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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
| 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, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Prime Difference Function
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, Sep 23 2008]
D. A. Goldston, J. Pintz, C. Y. Yildirim, Positive Proportion of Small Gaps Between Consecutive Primes, Mar 21, 2011.
D. A. Goldston, A. H. Ledoan, On the differences between consecutive prime numbers, I", arXiv:1111.3380v1 [math.NT], Nov 14, 2011 [Jonathan Vos Post, Nov 15, 2011]
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FORMULA
| G.f.: b(x)*(1-x), where b(x) is the g.f. for the primes. - Frank Adams-Watters, Jun 15 2006
a(n) = prime(n+1) - prime(n). [From Franklin T. Adams-Watters, Mar 31 2010]
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MAPLE
| with(numtheory): for n from 1 to 500 do printf(`%d, `, ithprime(n+1) - ithprime(n)) od:
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MATHEMATICA
| p = Table[Prime[i], {i, 1, 100}]; Drop[p, 1] - Drop[p, -1]
Array[ Mod[ Prime[ # + 1], Prime[ # ]] &, 97] [From Robert G. Wilson v, Jul 14 2010]
t = Array[Prime, 98]; Rest@t - Most@t [From Robert G. Wilson v, Jul 14 2010]
Differences[Prime[Range[100]]] (* From Harvey P. Dale, May 15 2011 *)
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PROG
| (Sage) differences(prime_range(1000)) # Joerg Arndt, May 15 2011.
(PARI) diff(v)=vector(#v-1, i, v[i+1]-v[i]);
diff(primes(100)) \\ Charles R Greathouse IV, Feb 11 2011
(MAGMA) [(NthPrime(n+1) - NthPrime(n)): n in [1..100]]; - Vincenzo Librandi, Apr 02 2011
(Haskell)
a001223 n = a001223_list !! (n-1)
a001223_list = zipWith (-) (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Oct 29 2011
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CROSSREFS
| Cf. A000040, A037201, A007921, A030173. Second difference is A036263, First occurrence is A000230.
Cf. A036263-A036274.
Sequence in context: A193562 A163824 A075526 * A171991 A118776 A092520
Adjacent sequences: A001220 A001221 A001222 * A001224 A001225 A001226
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001
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