

A001223


Differences between consecutive primes.
(Formerly M0296 N0108)


420



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;
text;
internal format)



OFFSET

1,2


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).  Rémi Eismann, Feb 14 2008
Shinya: Let p_{k} [A000040(k)] denote the kth 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 pape 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 ktuple 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]


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.
Beck, József. Inevitable randomness in discrete mathematics. University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 9780821847565; MR2543141 (2010m:60026). See page 7.
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 between prime numbers: the work of GoldstonPintzYildirim, Bull. Amer. Math. Soc., 44 (2007), 118.


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
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]
D. A. Goldston, J. Pintz, C. Y. Yildirim, Positive Proportion of Small Gaps Between Consecutive Primes, Mar 21, 2011.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959, 2014
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]
K. Soundararajan, Small gaps between prime numbers: the work of GoldstonPintzYildirim, Bull. Amer. Math. Soc., 44 (2007), 118.
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Prime Difference Function
Index entries for primes, gaps between


FORMULA

G.f.: b(x)*(1x), where b(x) is the g.f. for the primes.  Frank AdamsWatters, Jun 15 2006
a(n) = prime(n+1)  prime(n). [From Franklin T. AdamsWatters, Mar 31 2010]
Conjecture: a(n) = ceiling(prime(n)*(log(prime(n+1))log(prime(n)))).  Thomas Ordowski, Mar 19 2013
Conjecture: a(n) = floor(prime(n+1)*(log(prime(n+1))log(prime(n)))).  Thomas Ordowski, Mar 20 2013
Conjecture: a(n) = floor((prime(n)+prime(n+1))*(log(prime(n+1))log(prime(n)))/2).  Thomas Ordowski, Mar 21 2013


MAPLE

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


MATHEMATICA

p = Table[Prime[i], {i, 1, 100}]; Drop[p, 1]  Drop[p, 1]
Array[ Mod[ Prime[ # + 1], Prime[ # ]] &, 97] (* Robert G. Wilson v, Jul 14 2010 *)
t = Array[Prime, 98]; Rest@t  Most@t (* Robert G. Wilson v, Jul 14 2010 *)
Differences[Prime[Range[100]]] (* Harvey P. Dale, May 15 2011 *)


PROG

(Sage) differences(prime_range(1000)) # Joerg Arndt, May 15 2011.
(PARI) diff(v)=vector(#v1, 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 !! (n1)
a001223_list = zipWith () (tail a000040_list) a000040_list
 Reinhard Zumkeller, Oct 29 2011


CROSSREFS

Cf. A000040, A037201, A007921, A030173. Second difference is A036263, First occurrence is A000230.
For records see A005250, A005669.
Cf. A036263A036274.
Sequence in context: A082508 A193562 A075526 * A118776 A092520 A147848
Adjacent sequences: A001220 A001221 A001222 * A001224 A001225 A001226


KEYWORD

nonn,nice,easy,hear


AUTHOR

N. J. A. Sloane.


EXTENSIONS

More terms from James A. Sellers, Feb 19 2001


STATUS

approved



