A031131 -id:A031131 - cp's OEIS frontend

A001223 Prime gaps: differences between consecutive primes.

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

Offset: 1

N. J. A. Sloane

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

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.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 186-192.
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).

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.
Joel E. Cohen and Dexter Senft, Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often, Notes on Number Theory and Discrete Mathematics, Volume 31, Number 3, 494-503 (2025). See p. 495.
Péter L. Erdős, Gergely Harcos, Shubha R. Kharel, Péter Maga, Tamás Róbert Mezei and Zoltán Toroczkai, The sequence of prime gaps is graphic, Mathematische Annalen 388 (2024), 2195-2215.
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

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.

a001223 n = a001223_list !! (n-1)
a001223_list = zipWith (-) (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Oct 29 2011

[(NthPrime(n+1) - NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011

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

Differences[Prime[Range[100]]] (* Harvey P. Dale, May 15 2011 *)

diff(v)=vector(#v-1,i,v[i+1]-v[i]);
diff(primes(100)) \\ Charles R Greathouse IV, Feb 11 2011

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

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

differences(prime_range(1000)) # Joerg Arndt, May 15 2011

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

More terms from James Sellers, Feb 19 2001

A001043 Numbers that are the sum of 2 successive primes.

Original entry on oeis.org

5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508, 520

Offset: 1

N. J. A. Sloane, R. K. Guy

Arithmetic derivative (see A003415) of prime(n)*prime(n+1). - Giorgio Balzarotti, May 26 2011
A008472(a(n)) = A191583(n). - Reinhard Zumkeller, Jun 28 2011
With the exception of the first term, all terms are even. a(n) is divisible by 4 if the difference between prime(n) and prime(n + 1) is not divisible by 4; e.g., prime(n) = 1 mod 4 and prime(n + 1) = 3 mod 4. In general, for a(n) to be divisible by some even number m > 2 requires that prime(n + 1) - prime(n) not be a multiple of m. - Alonso del Arte, Jan 30 2012

			2 + 3 = 5.
3 + 5 = 8.
5 + 7 = 12.
7 + 11 = 18.

Archimedeans Problems Drive, Eureka, 26 (1963), 12.
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).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Albert Frank & Philippe Jacqueroux, International Contest, 2001. Item 22
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021)

Subsequence of A050936.

Cf. A000040 (primes), A031131 (first differences), A092163 (bisection), A100479 (bisection).

a001043 n = a001043_list !! (n-1)
a001043_list = zipWith (+) a000040_list $ tail a000040_list
-- Reinhard Zumkeller, Oct 19 2011

[(NthPrime(n+1) + NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011

Primes:= select(isprime,[2,seq(2*i+1,i=1..1000)]):
n:= nops(Primes):
Primes[1..n-1] + Primes[2..n]; # Robert Israel, Aug 29 2014

Table[Prime[n] + Prime[n + 1], {n, 55}] (* Ray Chandler, Feb 12 2005 *)
Total/@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Aug 23 2011 *)
Abs[Differences[Table[(-1)^n Prime[n], {n, 60}]]] (* Alonso del Arte, Feb 03 2016 *)

p=2;forprime(q=3,1e3,print1(p+q", ");p=q) \\ Charles R Greathouse IV, Jun 10 2011

is(n)=precprime((n-1)/2)+nextprime(n/2)==n&&n>2 \\ Charles R Greathouse IV, Jun 21 2012

BB = primes_first_n(56)
L = []
for i in range(55): L.append(BB[1 + i] + BB[i])
L # Zerinvary Lajos, May 14 2007

a(n) = prime(n) + prime(n + 1) = A000040(n) + A000040(n+1).
a(n) = A116366(n, n - 1) for n > 1. - Reinhard Zumkeller, Feb 06 2006
a(n) = 2*A024675(n-1), n>1. - R. J. Mathar, Jan 12 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

A046931 Prime islands: for n >= 2, a(n) = least prime whose adjacent primes are exactly 2n apart; a(1) = 3 by convention.

Original entry on oeis.org

3, 5, 7, 29, 23, 53, 89, 223, 113, 331, 631, 211, 1381, 1129, 1637, 4759, 2579, 3433, 4297, 1327, 2179, 2503, 7993, 5623, 9587, 17257, 15859, 14107, 19609, 34981, 36433, 33247, 24281, 35617, 43331, 19661, 134513, 31397, 137029

Offset: 1

Marc LeBrun and David W. Wilson

For n > 1: (n) = A000040(A261525(n)+1) for n > 1. - Reinhard Zumkeller, Aug 23 2015

			29 is in a sea of 6 composites, namely 24, 25, 26, 27, 28 and 30 and is the smallest such number, so a(4) = 29.

T. D. Noe, Table of n, a(n) for n=1..100
Abhimanyu Kumar and Anuraag Saxena, Insulated primes, arXiv:2011.14210 [math.NT], 2020. Mentions this sequence.

Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.
Another version: see A098968 and A098969.
Cf. A031131. - Zak Seidov, Jan 25 2015
Cf. A261525, A000040.

Join[{3},Transpose[Flatten[Table[Select[Partition[Prime[Range[ 20000]],3,1], Last[#]-First[#]==2 n&,1],{n,2,45}],1]][[2]]] (* Harvey P. Dale, May 22 2012 *)

Typo in example fixed by Zak Seidov, Jan 25 2015

A031165 a(n) = prime(n+3) - prime(n).

Original entry on oeis.org

5, 8, 8, 10, 8, 10, 12, 12, 14, 12, 12, 10, 12, 16, 14, 14, 12, 12, 12, 12, 16, 18, 18, 14, 10, 8, 10, 20, 22, 24, 12, 18, 14, 18, 14, 16, 16, 16, 14, 18, 14, 16, 8, 18, 26, 28, 18, 10, 12, 12, 18, 18, 22, 18, 14, 14, 12, 12, 16, 26, 28, 20, 10, 20, 24, 30, 18, 16, 12

Offset: 1

Jeff Burch, Dec 11 1999

nonn

Comments from Jonathan Vos Post, Jan 22 2006 (Start): This sequence is the k=3 case of the family of sequences a(k,n) = prime(n+k) - prime(n). See A001223 and A031131 for k = 1 and 2.

The records in this sequence give A115401. The minimal value, after the anomalous initial values (5, 8, 8), is 8 which occurs iff n is an element of A007530 (prime quadruples: numbers n such that n, n+2, n+6, n+8 are all prime). (End)

			a(1) = prime(4) - prime(1) = 7 - 2 = 5, which is the only odd element of this sequence.
a(2) = prime(5) - prime(2) = 11 - 3 = 8.
a(3) = prime(6) - prime(3) = 13 - 5 = 8.
a(4) = prime(7) - prime(4) = 17 - 7 = 10.
a(99) = prime(102) - prime(99) = 557 - 523 = 34. - _Jonathan Vos Post_, Jan 22 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A001223, A007530, A031131, A034961, A115401.

Cf. A031166, A031167, A031168, A031169, A031170, A031171, A031172.

a031165 n = a031165_list !! (n-1)
a031165_list = zipWith (-) (drop 3 a000040_list) a000040_list
-- Reinhard Zumkeller, Aug 23 2015

[NthPrime(n+3)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 11 2011

a:= n-> ithprime(n+3)-ithprime(n): seq (a(n), n=1..80);

t = Array[Prime, 75]; Drop[t, 3] - Drop[t, -3] (* Robert G. Wilson v *)
#[[4]]-#[[1]]&/@Partition[Prime[Range[80]],4,1] (* Harvey P. Dale, Nov 07 2021 *)

p=2;q=3;r=5;forprime(s=7,1e3,print1(s-p", "); p=q;q=r;r=s) \\ Charles R Greathouse IV, Nov 07 2012

a(n) = prime(n+3) - prime(n). a(n) = A000040(n+3) - A000040(n). - Jonathan Vos Post, Jan 22 2006

a(n) = A034961(n+1) - A034961(n). - Zak Seidov, Nov 07 2012

Edited by R. J. Mathar and N. J. A. Sloane, Aug 11 2008

A031172 a(n) = prime(n+10) - prime(n).

Original entry on oeis.org

29, 34, 36, 36, 36, 40, 42, 42, 44, 42, 42, 42, 42, 46, 50, 48, 44, 46, 42, 42, 54, 52, 54, 50, 52, 50, 54, 56, 58, 60, 52, 50, 54, 54, 48, 48, 54, 60, 60, 56, 54, 58, 50, 58, 60, 64, 58, 48, 50, 52, 50, 54, 66, 60, 56, 54, 62, 66, 70, 68, 70, 66, 60, 62, 66, 66, 58

Offset: 1

Jeff Burch

In principle, moderate values should appear infinitely many times, by analogy with twin primes hypothesis. For example, a(n) = 44 for n = 9, 17, 206, 1604467, 12905293, 18008874, 26545460, 32655424, 57848470, 58313630, 59022635, 66275281, 81581956, 123780499, 160884754, 167797255, 179786560, 181569324, 239542290, ... - Zak Seidov, Sep 14 2014, edited by M. F. Hasler, Dec 03 2018

According to the k-tuple conjecture, any admissible k-tuple of primes occurs with calculable nonzero asymptotic density, i.e., in particular, infinitely many times. For k = 11, number of primes in the interval [prime(n), prime(n+10)], the smallest possible diameter of a k-tuple is A008407(11) = 36, and there are A083409(11) = 2 such constellations: {0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36}, first occurring at A213646(1) = 1418575498573, and {0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36}, first occurring at A213647(1) = 11. The combined list { prime(n) | a(n) = 36 } is A257129. - M. F. Hasler, Dec 03 2018

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040.
Cf. A001223, A031131, A031165, A031166, A031167, A031168, A031169, A031170, A031171.
Cf. A008407, A083409, A213646, A213647, A257129.

P:=Filtered([1..400],IsPrime);; a:=List([1..Length(P)-10],n->P[n+10]-P[n]); # Muniru A Asiru, Dec 06 2018

a031172_list = zipWith (-) (drop 10 a000040_list) a000040_list
a031172 n = a031172_list !! (n-1)  -- Reinhard Zumkeller, Aug 23 2015

[NthPrime(n+10)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 23 2011

A031172:=n->ithprime(n+10)-ithprime(n): seq(A031172(n), n=1..50);

Table[Prime[n + 10] - Prime[n], {n, 50}] (* Wesley Ivan Hurt, Sep 14 2014 *)

A031172(n)=prime(n+10)-prime(n) \\ M. F. Hasler, Dec 03 2018

from sympy import prime
for n in range(1,100): print(prime(n+10)-prime(n)) # Stefano Spezia, Dec 06 2018

[(nth_prime(n+10) - nth_prime(n)) for n in (1..100)] # G. C. Greubel, Dec 04 2018

a(n) = A000040(n+10) - A000040(n). - Wesley Ivan Hurt, Sep 14 2014

Offset changed from 2 to 1; added a(1)=29 by Vincenzo Librandi, Apr 23 2011

cp's OEIS Frontend

A261525 a(n) = smallest m such that A031131(m) = 2*n.

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A033957 Duplicate of A031131.

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A001223 Prime gaps: differences between consecutive primes.

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A001043 Numbers that are the sum of 2 successive primes.

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A046931 Prime islands: for n >= 2, a(n) = least prime whose adjacent primes are exactly 2n apart; a(1) = 3 by convention.

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A031165 a(n) = prime(n+3) - prime(n).

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A031172 a(n) = prime(n+10) - prime(n).