A008407 -id:A008407 - cp's OEIS frontend

A083409 Number of prime k-tuplet constellations, i.e., patterns with minimal diameter A008407.

1, 2, 1, 2, 1, 2, 3, 4, 2, 2, 2, 6, 2, 4, 2, 4, 2, 4, 2, 2, 4, 2, 4, 18, 2, 8, 10, 2, 2, 2, 4, 14, 20, 2, 2, 2, 6, 26, 26, 8, 2, 6, 18, 4, 4, 4, 2, 2, 22, 22, 2, 2, 26, 6, 6, 2, 2, 4, 2, 2, 6, 2, 2, 2, 2, 18, 2, 20, 2, 2, 2, 10, 2, 14, 14, 40, 8, 2, 14, 14, 16, 4, 2, 2, 60, 50, 2, 2, 2, 16, 2, 18, 12

Offset: 2

Frank Ellermann, Jun 07 2003

nonn

			For a(8) = 3 octuplet patterns see A065706. for a(6) = 1 sextet see A061671.

T. D. Noe, Table of n, a(n) for n=2..305 (from Engelsma's data)
Thomas J Engelsma, Permissible Patterns.
Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A008407, A020497, A023193, A061671, A065688, A065706.

More terms from Engelsma's website sent by T. D. Noe, Jul 21 2006

A261324 Smallest prime p that starts an n-tuplet of consecutive primes of length A008407(n).

Original entry on oeis.org

2, 3, 5, 3, 5, 7, 11, 11, 7, 5, 5, 11, 11, 11, 11, 7, 13, 13, 13, 29, 29, 7, 7

Offset: 1

Max Alekseyev, Aug 14 2015

In contrast to A065688, n-tuplets here may be singular and give the complete set of residues modulo some prime. For example, for n=4 we have the 4-tuplet: (3,5,7,11) = (3,3+2,3+4,3+8), but there are no other prime 4-tuplets of the form (p,p+2,p+4,p+8), since one of its elements would be divisible by 3.

For any n, a(n) <= n or a(n) = A065688(n).

Cf. A000040, A008407, A065688, A261323.

a(n) = A000040(A261323(n)).

A261323 Smallest m such that prime(m+n-1) - prime(m) = A008407(n); that is, prime(m) starts the smallest n-tuplet of consecutive primes of length A008407(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 5, 4, 3, 3, 5, 5, 5, 5, 4, 6, 6, 6, 10, 10, 4, 4

Offset: 1

Max Alekseyev, Aug 14 2015

See A261324 for further comments and the relation to A065688.

Cf. A008407, A261324.

A261323(n,d=A008407[n],m=0)={until(prime(m+n)==prime(m++)+d,);m} \\ Assumes a precomputed vector A008407 with at least n elements, or supply the gap as 2nd arg. Inefficient for n>23. - M. F. Hasler, Aug 17 2015

a(n) = A000720(A261324(n)). - M. F. Hasler, Aug 17 2015

A227083 Number of ways to write n as a + b/2 with a and b terms of the sequence A008407.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 4, 1, 4, 4, 2, 3, 5, 3, 4, 5, 4, 3, 7, 4, 4, 3, 6, 5, 5, 3, 6, 5, 6, 4, 6, 4, 6, 7, 5, 5, 7, 4, 6, 6, 7, 4, 7, 6, 5, 8, 5, 6, 9, 6, 5, 6, 7, 8, 8, 6, 7, 7, 9, 7, 7, 5, 9, 10, 6, 8, 9, 8, 10, 7, 8, 7, 11, 8, 7, 9, 9, 10, 10, 8, 9, 8, 13

Offset: 1

Zhi-Wei Sun, Jun 30 2013

nonn

Conjecture: We have a(n) > 0 for all n > 4.

For every k = 2, ..., 342, the value of A008407(k) has been determined by T. J. Engelsma. Since A008407(343)/2 > A008407(342)/2 = 2328/2 = 1164, if n <= 1166 can be written as A008407(j) + A008407(k)/2 with j > 1 and k > 1 then neither j nor k exceeds 342. Based on this we are able to compute a(n) for n = 1, ..., 1166.

			a(10) = 2 since 10 = 2 + 16/2 = 6 + 8/2;
a(11) = 1 since 11 = 8 + 6/2;
a(25) = 1 since 25 = 12 + 26/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1166
A. V. Sutherland, Narrow admissible k-tuples: bounds on H(k), 2013.
T. Tao, Bounded gaps between primes, PolyMath Wiki Project, 2013.

Cf. A008407.

A227156 Number of ways to write n as a sum of a square and half of a term of the sequence A008407.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 4, 1, 4, 2, 1, 4, 0, 3, 4, 3, 1, 3, 4, 2, 3, 2, 3, 4, 2, 1, 4, 2, 4, 4, 3, 4, 2, 4, 1, 5, 2, 1, 5, 3, 5, 2, 2, 4, 3, 3, 3, 3, 3, 3, 4, 1, 3, 5, 4, 4, 4, 2, 4, 4, 3, 4, 3, 6, 1, 3, 5, 2, 6, 4, 3, 7, 2, 4, 6, 2, 3, 5, 5, 2, 4, 4, 4, 6, 4, 3, 5, 3, 5, 3

Offset: 1

Zhi-Wei Sun, Jul 02 2013

nonn

Conjecture: We have a(n) > 0 except for n = 23.

We also conjecture that any positive integer can be written as a sum of a triangular number and half of a term of A008407, and each integer n > 4 can be written as x + y (x>0, y>0) with x*y a term of A008407.

			a(195) = 1 since 195 = 0^2 + A008407(23)/2.
a(378) = 1 since 378 = 8^2 + A008407(110)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1164
A. V. Sutherland, Narrow admissible k-tuples: bounds on H(k), 2013.
T. Tao, Bounded gaps between primes, PolyMath Wiki Project, 2013.

Cf. A000290, A008407, A227083.

A007918 Least prime >= n (version 1 of the "next prime" function).

Original entry on oeis.org

2, 2, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71, 71, 71, 71, 73, 73

Offset: 0

R. Muller and Charles T. Le (charlestle(AT)yahoo.com)

Version 2 of the "next prime" function is "smallest prime > n". This produces A151800.
Maple uses version 2.
According to the "k-tuple" conjecture, a(n) is the initial term of the lexicographically earliest increasing arithmetic progression of n primes; the corresponding common differences are given by A061558. - David W. Wilson, Sep 22 2007
It is easy to show that the initial term of an increasing arithmetic progression of n primes cannot be smaller than a(n). - N. J. A. Sloane, Oct 18 2007
Also, smallest prime bounded by n and 2n inclusively (in accordance with Bertrand's theorem). Smallest prime >n is a(n+1) and is equivalent to smallest prime between n and 2n exclusively. - Lekraj Beedassy, Jan 01 2007
Run lengths of successive equal terms are given by A125266. - Felix Fröhlich, May 29 2022
Conjecture: if n > 1, then a(n) < n^(n^(1/n)). - Thomas Ordowski, Feb 23 2023

T. D. Noe, Table of n, a(n) for n = 0..10000
Jens Kruse Andersen, Records for primes in arithmetic progressions
K. Atanassov, On Some of Smarandache's Problems
K. Atanassov, On the 37th and 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
Henry Bottomley, Prime number calculator
J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.
Andrew Granville, Prime Number Patterns
Hans Gunter, Puzzle 145. The Inferior Smarandache Prime Part and Superior Smarandache Prime Part functions; Solutions by Jean Marie Charrier, Teresinha DaCosta, Rene Blanch, Richard Kelley and Jim Howell.
Jonathan Sondow and Eric Weisstein, Bertrand's Postulate, World of Mathematics.
Eric Weisstein's World of Mathematics, Next Prime, k-tuple conjecture
Index entries for sequences related to primes in arithmetic progressions

Cf. A000040, A007917, A008407, A020497, A061558, A125266, A151799, A151800, A171400.

a007918 n = a007918_list !! n
a007918_list = 2 : 2 : 2 : concat (zipWith
              (\p q -> (replicate (fromInteger(q - p)) q))
                                   a000040_list $ tail a000040_list)
-- Reinhard Zumkeller, Jul 26 2012

[2] cat [NextPrime(n-1): n in [1..80]]; // Vincenzo Librandi, Jan 14 2016

A007918 := n-> nextprime(n-1); # M. F. Hasler, Apr 09 2008

NextPrime[Range[-1, 72]] (* Jean-François Alcover, Apr 18 2011 *)

A007918(n)=nextprime(n)  \\ M. F. Hasler, Jun 24 2011

for(x=0,100,print1(nextprime(x)",")) \\ Cino Hilliard, Jan 15 2007

from sympy import nextprime
def A007918(n): return nextprime(n-1) # Chai Wah Wu, Apr 22 2022

For n > 1: a(n) = A000040(A049084(A007917(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012

a(n) = A151800(n-1). - Seiichi Manyama, Apr 02 2018

A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

Original entry on oeis.org

11, 17, 1277, 88793, 113147, 284723, 855713, 1146773, 2580647, 6560993, 15760091, 20737877, 25658441, 58208387, 69156533, 73373537, 74266253, 76170527, 93625991, 100658627, 134764997, 137943347, 165531257, 171958667

Offset: 1

Frank Ellermann, Dec 05 2001

nonn

3 patterns for 8-tuplets: 11010011001011, 11011010011001 and v.v.

See A022011, A022012 and A022013 for the three different possible patterns. The sequence is conjectured to be infinite, although it is not even proved that there are infinitely many twin primes (p1, p2 = p1+2). - M. F. Hasler, May 02 2015

			a(3) = 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303 = 1277+26 are primes.

Harry J. Smith and Dana Jacobsen, Table of n, a(n) for n = 1..18123 [first 100 terms from Harry J. Smith]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, The smallest prime k-tuplets, database of compressed files.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

11 = A065688(8), 26 = A008407(8), 8 = A023193(26+1), octets in A066082 are another (not minimal) constellation of 8 primes.
Union of A022011, A022012 and A022013.
See A257124 (prime septuplets) with an overview of prime k-tuplets.

{ n=0; p1=2; p8=19; for (m=1, 10^12, p1=nextprime(p1+1); p8=nextprime(p8+1); if (p8 - p1 == 26, write("b065706.txt", n++, " ", p1); if (n==100, return)) ) } \\ Harry J. Smith, Oct 26 2009

use ntheory ":all"; my($s,$e,$i,%h)=(1,1e10,0); undef @h{sieve_prime_cluster($s,$e,2,6,8,12,18,20,26), sieve_prime_cluster($s,$e,2,6,12,14,20,24,26), sieve_prime_cluster($s,$e,6,8,14,18,20,24,26)}; say ++$i," $" for sort {$a<=>$b} keys %h; # _Dana Jacobsen, Oct 10 2015

A257129 Initial members of prime 11-tuples.

Original entry on oeis.org

11, 1418575498573, 2118274828903, 4396774576273, 6368171154193, 6953798916913, 7908189600581, 10527733922591, 12640876669691, 27899359258003, 28138953913303, 34460918582323, 38545620633251, 40362095929003, 42023308245613, 43564522846961, 44058461657443, 60268613366231, 60596839933361, 61062361183903, 71431649320301

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

It appears that this lists only starting primes for one of the A083409(11) = 2 constellations with minimal diameter A008407(11) = 36, i.e., the union of A213646 and A213647, while there are other prime 11-tuples with larger diameter. - M. F. Hasler, Dec 03 2018

Tim Johannes Ohrtmann and Dana Jacobsen, Table of n, a(n) for n = 1..13723 [This replaces an earlier b-file from Tim Johannes Ohrtmann]
Tony Forbes k-tuplets

Initial members of all of the first prime k-tuples:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruples: A007530.
prime quintuples: A086140 out of A022007, A022006.
prime sextuples: A022008.
prime septuples: A257124 out of A022009, A022010.
prime octuples: A065706 out of A022011, A022012, A022013.
prime nonuples: A257125 out of A022547, A022548, A022545, A022546.
prime 10-tuples: A257127 out of A027569, A027570.
prime 11-tuples: this sequence out of A213646, A213647.
prime 12-tuples: A257131 out of A213601, A213645.
prime 13-tuples: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuples: A257166 out of A257167, A257168.
prime 15-tuples: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuples: A257308 out of A257369, A257370.
prime 17-tuples: A257373 out of A257374, A257375, A257376, A257377.

A020497 Conjecturally, this is the minimal y such that n primes occur infinitely often among (x+1, ..., x+y), that is, pi(x+y) - pi(x) >= n for infinitely many x.

Original entry on oeis.org

1, 3, 7, 9, 13, 17, 21, 27, 31, 33, 37, 43, 49, 51, 57, 61, 67, 71, 77, 81, 85, 91, 95, 101, 111, 115, 121, 127, 131, 137, 141, 147, 153, 157, 159, 163, 169, 177, 183, 187, 189, 197, 201, 211, 213, 217, 227, 237, 241, 247, 253, 255, 265, 271, 273, 279, 283, 289, 301, 305

Offset: 1

Robert G. Wilson v, Christopher E. Thompson

a(n) purportedly gives the least k with A023193(k) = n; that is, this sequence should be the "least inverse" of A023193.
My web page extends the sequence to rho(305)=2047 and also gives a super-dense occurrence at rho(592)=4333 when pi(4333)=591 - the first known occurrence. - Thomas J Engelsma (tom(AT)opertech.com), Feb 16 2004
Tomás Oliveira e Silva (see link) has a table extending to n = 1000.
The minimal y such that there are n elements of {1, ..., y} with fewer than p distinct elements mod p for all prime p. - Charles R Greathouse IV, Jun 13 2013

R. K. Guy, Unsolved Problems in Number Theory, (2nd edition, Springer, 1994), Section A9.

T. D. Noe, Table of n, a(n) for n = 1..672 (from Engelsma's data)
Thomas J. Engelsma, Permissible Patterns.
T. Forbes, Prime k-tuplets
Daniel M. Gordon and Gene Rodemich, Dense admissible sets, Proceedings of ANTS III, LNCS 1423 (1998), pp. 216-225.
D. Hensley and I. Richards, Primes in intervals, Acta Arith. 25 (1974), pp. 375-391.
H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), pp. 119-134.
Tomás Oliveira e Silva, Admissible prime constellations
Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bulletin of the American Mathematical Society 80:3 (1974), pp. 419-438.
H. Smith, On a generalization of the prime pair problem, Math. Comp., 11 (1957) 249-254.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Equals A008407 + 1. First differences give A047947.

Cf. A023193 (prime k-tuplet conjectures), A066081 (weaker binary conjectures).

Prime(floor((n+1)/2)) <= a(n) < prime(n) for large n. See Hensley & Richards and Montgomery & Vaughan. - Charles R Greathouse IV, Jun 18 2013

Corrected and extended by David W. Wilson

A113274 Record gaps between twin primes.

Original entry on oeis.org

2, 6, 12, 18, 30, 36, 72, 150, 168, 210, 282, 372, 498, 630, 924, 930, 1008, 1452, 1512, 1530, 1722, 1902, 2190, 2256, 2832, 2868, 3012, 3102, 3180, 3480, 3804, 4770, 5292, 6030, 6282, 6474, 6552, 6648, 7050, 7980, 8040, 8994, 9312, 9318, 10200, 10338, 10668

Offset: 1

Bernardo Boncompagni, Oct 21 2005

nonn

a(n) mod 6 = 0 for each n>1.

			The first twin primes are 3,5 and 5,7 so a(0)=5-3=2. The following pair is 11,13 so a(1)=11-5=6. The following pair is 17,19 so 6 remains the record and no terms are added.

Martin Raab, Table of n, a(n) for n = 1..82 (terms up to a(72) from Alexei Kourbatov, a(73)-a(75) from Tomás Oliveira e Silva)
Richard Fischer, Maximale Lücken (Intervallen) von Primzahlenzwillingen
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013; and J. Int. Seq. 16 (2013) #13.5.2
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.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Mersenneforum, Gaps between prime pairs (Twin Primes).
Tomás Oliveira e Silva, Gaps between twin primes
Luis Rodriguez and Carlos Rivera, Conjecture 66. Gaps between consecutive twin pairs, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
Eric Weisstein's World of Mathematics, Twin Prime Constant

The smallest primes originating the sequence are given in A113275. Cf. A008407, A005250, A002386.

NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k+=6]; k]; p = 5; r = 2; t = {2}; Do[ q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, q - p]; Print[{p, q - p}]; r = q - p]; p = q, {n, 10^9}]; t (* Robert G. Wilson v, Oct 22 2005 *)
DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],2,1],#[[2]]-#[[1]] == 2&][[All,2]]],GreaterEqual] (* The program generates the first 27 terms of the sequence. *) (* Harvey P. Dale, Dec 31 2022 *)

a(n) = A036063(n) + 2.
a(n) = A036062(n) - A113275(n).
From Alexei Kourbatov, Dec 29 2011: (Start)
(1) Upper bound: gaps between twin primes are smaller than 0.76*(log p)^3, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)-1.2), where a = 0.76*(log p)^2 is the average gap between twin primes near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
(End)

More terms from Robert G. Wilson v, Oct 22 2005
Corrected terms based on A036063, cross-checked with independent computations by Carlos Rivera and Richard Fischer (linked).
Terms up to a(72) are given in Kourbatov (2013), terms up to a(75) in Oliveira e Silva website.

cp's OEIS Frontend

A083409 Number of prime k-tuplet constellations, i.e., patterns with minimal diameter A008407.

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A261324 Smallest prime p that starts an n-tuplet of consecutive primes of length A008407(n).

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A261323 Smallest m such that prime(m+n-1) - prime(m) = A008407(n); that is, prime(m) starts the smallest n-tuplet of consecutive primes of length A008407(n).

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A227083 Number of ways to write n as a + b/2 with a and b terms of the sequence A008407.

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A227156 Number of ways to write n as a sum of a square and half of a term of the sequence A008407.

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A007918 Least prime >= n (version 1 of the "next prime" function).

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A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

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A257129 Initial members of prime 11-tuples.

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A020497 Conjecturally, this is the minimal y such that n primes occur infinitely often among (x+1, ..., x+y), that is, pi(x+y) - pi(x) >= n for infinitely many x.

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