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 A001223 Prime gaps: differences between consecutive primes. (Formerly M0296 N0108) 687
 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 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 -> oo} b(n,j,k)/b(n,j,1/k) = 1, for any j > 0 and k > 0, and lim_{n -> oo} 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. GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92. 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 Carlos Rivera, Conjecture 82. Average of log Dn / log(logPn) equal R = 0,877 08..., The Prime Puzzles & Problems Connection. 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 Yasuo Yamasaki 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 Conjectures: (i) a(n) = ceiling(prime(n)*log(prime(n+1)/prime(n))). (ii) a(n) = floor(prime(n+1)*log(prime(n+1)/prime(n))). (iii) 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 Conjecture: Limit_{N->oo} (Sum_{n=2..N} log(a(n))) / (Sum_{n=2..N} log(log(prime(n)))) = 1. - Alain Rocchelli, Dec 16 2022 Conjecture: The asymptotic limit of the average of log(a(n)) ~ log(log(prime(n))) - gamma (where gamma is Euler's constant). Also, for n tending to infinity, the geometric mean of a(n) is equivalent to log(prime(n)) / e^gamma. - Alain Rocchelli, Jan 23 2023 It has been conjectured that primes are distributed around their average spacing in a Poisson distribution (cf. D. A. Goldston in above links). This is the basis of the last two conjectures above. - Alain Rocchelli, Feb 10 2023 MAPLE with(numtheory): for n from 1 to 500 do printf(`%d, `, ithprime(n+1) - ithprime(n)) od: MATHEMATICA 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(#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,changed AUTHOR N. J. A. Sloane EXTENSIONS More terms from James A. Sellers, Feb 19 2001 STATUS approved

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