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A001223 Prime gaps: differences between consecutive primes.
(Formerly M0296 N0108)
658
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 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

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

A solution (modular square root) of x^2 == A001248(n) (mod A000040(n+1)). - L. Edson Jeffery, Oct 01 2014

There exists a constant C such that for n -> infinity, Cramer conjecture a(n) < C log^2 prime(n) is equivalent to (log prime(n+1)/log prime(n))^n < e^C. - Thomas Ordowski, Oct 11 2014

a(n) = A008347(n+1) - A008347(n-1). - Reinhard Zumkeller, Feb 09 2015

Yitang Zhang proved lim inf_{n -> infinity} a(n) is finite. - Robert Israel, Feb 12 2015

lim sup_{n -> infinity} a(n)/log^2 prime(n) = C <==> lim sup_{n -> infinity}(log prime(n+1)/log prime(n))^n = e^C. - Thomas Ordowski, Mar 09 2015

a(A038664(n)) = 2*n and a(m) != 2*n for m < A038664(n). - Reinhard Zumkeller, Aug 23 2015

If j and k are positive integers then there are no two consecutive primes gaps of the form 2+6j and 2+6k (A016933) or 4+6j and 4+6k (A016957). - Andres Cicuttin, Jul 14 2016

Conjecture: For any positive numbers x and y, there is an index k such that x/y = a(k)/a(k+1). - Andres Cicuttin, Sep 23 2018

Conjecture: For any three positive numbers x, y and j, there is an index k such that x/y = a(k)/a(k+j). - Andres Cicuttin, Sep 29 2018

Conjecture: For any three positive numbers x, y and j, there are infinitely many indices k such that x/y = a(k)/a(k+j). - Andres Cicuttin, Sep 29 2018

Row m of A174349 lists all indices n for which a(n) = 2m. - M. F. Hasler, Oct 26 2018

Since (6a, 6b) is an admissible pattern of gaps for any integers a, b > 0 (and also if other multiples of 6 are inserted in between), the above conjecture follows from the prime k-tuple conjecture which states that any admissible pattern occurs infinitely often (see, e.g., the Caldwell link). This also means that any subsequence a(n .. n+m) with n > 2 (as to exclude the untypical primes 2 and 3) should occur infinitely many times at other starting points n'. - M. F. Hasler, Oct 26 2018

Conjecture: Defining b(n,j,k) as the number of pairs of prime gaps {a(i),a(i+j)} such that i<n, j>0, and a(i)/a(i+j)=k with k>0, then

  lim_{n -> infinity} b(n,j,k)/b(n,j,1/k) = 1, for any j>0 and k>0, and

  lim_{n -> infinity} b(n,j,k1)/b(n,j,k2) = C with C = C(j,k1,k2)>0. - Andres Cicuttin, Sep 01 2019

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).

LINKS

Vojtech Strnad, First 100000 terms [First 10000 terms from N. J. A. Sloane]

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].

Anonymous ["TheHereticAnthem20"], Prime gaps mapped to sounds, Youtube video (2018).

B. Apostol, L. Panaitopol, L Petrescu, and L. Toth, Some Properties of a Sequence Defined with the Aid of Prime Numbers, J. Int. Seq. 18 (2015) # 15.5.5.

S. Ares and M. Castro, Hidden structure in the randomness of the prime number sequence?, arXiv:cond-mat/0310148 [cond-mat.stat-mech], 2003-2005.

József Beck, Inevitable randomness in discrete mathematics, University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 7.

Chris K. Caldwell, Prime k-tuple conjecture, Prime Pages' Glossary entry.

D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between primes and almost primes, arXiv:math/0506067 [math.NT], 2005.

D. A. Goldston and A. H. Ledoan, On the differences between consecutive prime numbers, I", arXiv:1111.3380v1 [math.NT], Nov 14, 2011.

D. A. Goldston, J. Pintz, and C. Y. Yildirim, Positive Proportion of Small Gaps Between Consecutive Primes, arXiv:1103.3986 [math.NT], Mar 21, 2011.

D. R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Bull. Amer. Math. Soc. 1 (1979), 758-760.

Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.

Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.

The Polymath project, Bounded gaps between primes

Hisanobu Shinya, On the density of prime differences less than a given magnitude which satisfy a certain inequality, arXiv:0809.3458 [math.GM], 2008-2011.

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

Eric Weisstein's World of Mathematics, Andrica's Conjecture

Eric Weisstein's World of Mathematics, Prime Difference Function

Yamasaki, Yasuo, and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).

Yitang Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), 1121-1174.

Index entries for primes, gaps between

FORMULA

G.f.: b(x)*(1-x), where b(x) is the g.f. for the primes. - Franklin T. Adams-Watters, Jun 15 2006

a(n) = prime(n+1) - prime(n). - Franklin T. Adams-Watters, Mar 31 2010

Conjecture: a(n) = ceiling(prime(n)*log(prime(n+1)/prime(n))). - Thomas Ordowski, Mar 19 2013

Conjecture: a(n) = floor(prime(n+1)*log(prime(n+1)/prime(n))). - Thomas Ordowski, Mar 20 2013

Conjecture: a(n) = floor((prime(n)+prime(n+1))*log(prime(n+1)/prime(n))/2). - Thomas Ordowski, Mar 21 2013

A167770(n) == a(n)^2 (mod A000040(n+1)). - L. Edson Jeffery, Oct 01 2014

a(n) = Sum_{k=1..2^(n+1)-1} (floor(cos^2(Pi*(n+1)^(1/(n+1))/(1+primepi(k))^(1/(n+1))))). - Anthony Browne, May 11 2016

G.f.: (Sum_{k>=1} x^pi(k)) - 1, where pi(k) is the prime counting function. - Benedict W. J. Irwin, Jun 13 2016

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 *)

a[n_] := PowerMod[Prime[n]^2, 1/2, Prime[n + 1]]; Table[a[n], {n, 97}] (* L. Edson Jeffery, Oct 01 2014 *)

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

(PARI) forprime(p=1, 1e3, print1(nextprime(p+1)-p, ", ")) \\ Felix Fröhlich, Sep 06 2014

(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

(Python)

from sympy import prime

def A001223(n): return prime(n+1)-prime(n) # Chai Wah Wu, Jul 07 2022

CROSSREFS

Cf. A000040 (primes), A001248 (primes squared), A000720, A037201, A007921, A030173, A036263-A036274, A167770, A008347.

Second difference is A036263, first occurrence is A000230.

For records see A005250, A005669.

Cf. A038664, A031131, A031165, A031166, A031167, A031168, A031169, A031170, A031171, A031172.

Cf. A174349, A029707, A029709, A320701, ..., A320720.

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556-A330561.

Sequence in context: A249868 A255311 A075526 * A118776 A249867 A351612

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

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Last modified October 6 04:09 EDT 2022. Contains 357261 sequences. (Running on oeis4.)