A045881 -id:A045881 - cp's OEIS frontend

A000230 a(0)=2; for n>=1, a(n) = smallest prime p such that there is a gap of exactly 2n between p and next prime, or -1 if no such prime exists.

2, 3, 7, 23, 89, 139, 199, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, 5591, 1327, 9551, 30593, 19333, 16141, 15683, 81463, 28229, 31907, 19609, 35617, 82073, 44293, 43331, 34061, 89689, 162143, 134513, 173359, 31397, 404597, 212701, 188029, 542603, 265621, 461717, 155921, 544279, 404851, 927869, 1100977, 360653, 604073

Offset: 0

N. J. A. Sloane

p + 1 = A045881(n) starts the smallest run of exactly 2n-1 successive composite numbers. - Lekraj Beedassy, Apr 23 2010

Weintraub gives upper bounds on a(252), a(255), a(264), a(273), and a(327) based on a search from 1.1 * 10^16 to 1.1 * 10^16 + 1.5 * 10^9, probably performed on a 1970s microcomputer. - Charles R Greathouse IV, Aug 26 2022

			The following table, based on a very much larger table in the web page of Tomás Oliveira e Silva (see link) shows, for each gap g, P(g) = the smallest prime such that P(g)+g is the smallest prime number larger than P(g);
* marks a record-holder: g is a record-holder if P(g') > P(g) for all (even) g' > g, i.e., if all prime gaps are smaller than g for all primes smaller than P(g); P(g) is a record-holder if P(g') < P(g) for all (even) g' < g.
This table gives rise to many sequences: P(g) is A000230, the present sequence; P(g)* is A133430; the positions of the *'s in the P(g) column give A100180, A133430; g* is A005250; P(g*) is A002386; etc.
   -----
   g P(g)
   -----
   1* 2*
   2* 3*
   4* 7*
   6* 23*
   8* 89*
   10 139*
   12 199*
   14* 113
   16 1831*
   18* 523
   20* 887
   22* 1129
   24 1669
   26 2477*
   28 2971*
   30 4297*
   32 5591*
   34* 1327
   36* 9551*
   ........
The first time a gap of 4 occurs between primes is between 7 and 11, so a(2)=7 and A001632(2)=11.

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).

Brian Kehrig, Table of n, a(n) for n = 0..721 (terms 0..672 from Hugo Pfoertner)
A. Booker, The Nth Prime Page
L. J. Lander and T. R. Parkin, On the first appearance of prime differences, Math. Comp., 21 (1967), 483-488.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Tomás Oliveira e Silva, Gaps between consecutive primes
J. Thonnard, Les nombres premiers (Primality check; Closest next prime; Factorizer)
Sol Weintraub, A large prime gap, Mathematics of Computation Vol. 36, No. 153 (Jan 1981), p. 279.
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Volume 179 (2014), Issue 3, pp. 1121-1174.
Index entries for primes, gaps between

A001632(n) = 2n + a(n) = nextprime(a(n)).
Cf. A001223, A002386, A005250, A045881, A038664.
Cf. A100964 (least prime number that begins a prime gap of at least 2n).
Cf. also A133429 (records), A133430, A100180, A226657, A229021, A229028, A229030, A229033, A229034.

Join[{2}, With[{pr = Partition[Prime[Range[86000]], 2, 1]}, Transpose[ Flatten[ Table[Select[pr, #[[2]] - #[[1]] == 2n &, 1], {n, 50}], 1]][[1]]]] (* Harvey P. Dale, Apr 20 2012 *)

a(n)=my(p=2);forprime(q=3,,if(q-p==2*n,return(p));p=q) \\ Charles R Greathouse IV, Nov 20 2012

use ntheory ":all"; my($l,$i,@g)=(2,0); forprimes { $g[($-$l) >> 1] //= $l;  while (defined $g[$i]) { print "$i $g[$i]\n"; $i++; }  $l=$; } 1e10; # Dana Jacobsen, Mar 29 2019

import numpy
from sympy import sieve as prime
aupto = 50
A000230 = np.zeros(aupto+1, dtype=object)
A000230[0], it = 2, 2
while all(A000230) == 0:
    gap = (prime[it+1] - prime[it]) // 2
    if gap <= aupto and A000230[gap] == 0: A000230[gap] = prime[it]
    it += 1
print(list(A000230)) # Karl-Heinz Hofmann, Jun 07 2023

a(n) = A000040(A038664(n)). - Lekraj Beedassy, Sep 09 2006

a(29)-a(37) from Jud McCranie, Dec 11 1999
a(38)-a(49) from Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 11 2002
"or -1 if ..." added to definition at the suggestion of Alexander Wajnberg by N. J. A. Sloane, Feb 02 2020

A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1.

Original entry on oeis.org

1, 3, 8, 23, 33, 45, 29, 281, 98, 153, 188, 262, 366, 428, 589, 737, 216, 1182, 3301, 2190, 1878, 1830, 7969, 3076, 3426, 2224, 3792, 8027, 4611, 4521, 3643, 8687, 14861, 12541, 15782, 3384, 34201, 19025, 17005, 44772, 23282, 38589, 14356

Offset: 1

Robert G. Wilson v, Aug 15 2002

nonn

Also, least k such that 2n = A001223(k-1) = prime(k+1) - prime(k), where prime(k) = A001223(n). - Alexander Adamchuk, Jul 30 2006

Also the least number k>0 such that the k-th maximal run of composite numbers has length 2n-1. For example, the 8th such run (24,25,26,27,28) is the first of length 2(3)-1, so a(3) = 8. Also positions of first appearances in A176246 (A046933 without first term). - Gus Wiseman, Jun 12 2024

			a(3) = 8 because 1+0+2-2+2-2+2+2 = 5 and (5+1)/2 = 3.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A000230, A001223.
Position of first appearance of 2n+1 in A176246.
For nonsquarefree runs we have a bisection of A373199.
A000040 lists the primes, first differences A001223.
A002808 lists the composite numbers, differences A073783, sums A053767.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005381, A027833, A038664, A045881, A068780, A174965, A371201, A373403.

NextPrim[n_Integer] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {50}]; s = 0; k = 1; p = 0; q = 2; r = 3; While[k < 10^6, p = q; q = r; r = NextPrim[q]; s = s + p + r - 2q; If[s < 101 && a[[(s + 1)/2]] == 0, a[[(s + 1)/2]] = k]; k++ ]; a

a001223(n) = prime(n+1) - prime(n);
a(n) = {my(k = 1); while(2*n != A001223(k+1), k++); k;} \\ Michel Marcus, Nov 20 2016

a(n) = A038664(n) - 1. - Filip Zaludek, Nov 19 2016

A271211 Composite integers sandwiched between primes p, q with q-p = 4.

Original entry on oeis.org

8, 9, 10, 14, 15, 16, 20, 21, 22, 38, 39, 40, 44, 45, 46, 68, 69, 70, 80, 81, 82, 98, 99, 100, 104, 105, 106, 110, 111, 112, 128, 129, 130, 164, 165, 166, 194, 195, 196, 224, 225, 226, 230, 231, 232, 278, 279, 280, 308, 309, 310, 314, 315, 316, 350, 351, 352, 380

Offset: 1

Michel Marcus, Apr 02 2016

nonn

			The composite number 8 is sandwiched between primes 7 and 11, and 11-7=4, so 8 is a member of the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Laurent Coppey, Décompositions multiplicatives directes des entiers, Diagrammes, 65-66 (2011), p. 1-68, in French, see J4 p. 11.

Cf. A014574, A029710, A045881.

Range[#[[1]]+1,#[[2]]-1]&/@Select[Partition[Prime[Range[100]],2,1],#[[2]]- #[[1]] == 4&]//Flatten (* Harvey P. Dale, Oct 12 2019 *)

lista(nn) = {forcomposite(c=4, nn, if ((p=precprime(c)) && ((nextprime(c)-p)==4), print1(c, ", ")););}

A075067 Start of the first run of exactly n consecutive odd composite numbers.

Original entry on oeis.org

9, 25, 91, 141, 201, 115, 1833, 525, 889, 1131, 1671, 2479, 2973, 4299, 5593, 1329, 9553, 30595, 19335, 16143, 15685, 81465, 28231, 31909, 19611, 35619, 82075, 44295, 43333, 34063, 89691, 162145, 134515, 173361, 31399, 404599, 212703

Offset: 1

Amarnath Murthy, Sep 08 2002

nonn

More precisely, a(n)-2 and a(n) + 2n are two consecutive primes and are the smallest successive prime pair with a difference 2(n+1).

This is a trivial variation of A000230 and A045881.

See A189118 for another version.

More terms from Matthew Conroy, Sep 10 2002

A271232 Composite integers sandwiched between primes p, q with q-p = 6.

Original entry on oeis.org

24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 74, 75, 76, 77, 78, 84, 85, 86, 87, 88, 132, 133, 134, 135, 136, 152, 153, 154, 155, 156, 158, 159, 160, 161, 162, 168, 169, 170, 171, 172, 174, 175, 176, 177, 178

Offset: 1

Michel Marcus, Apr 02 2016

nonn

			The composite number 24 is sandwiched between consecutive primes 23 and 29, and 29-23=6, so 24 is a member of the sequence.

Laurent Coppey, Décompositions multiplicatives directes des entiers, Diagrammes, 65-66 (2011), p. 1-68, in French, see J6 p. 11.

Cf. A014574, A031924, A031925, A045881, A271211.

lista(nn) = {forcomposite(c=4, nn, if ((p=precprime(c)) && ((nextprime(c)-p)==6), print1(c, ", ")););}

A340735 a(n) is the smallest positive integer that begins a run of exactly 2*n-1 consecutive integers having at least 4 divisors each.

Original entry on oeis.org

6, 14, 32, 90, 140, 200, 294, 1832, 1070, 888, 1130, 2180, 2478, 2972, 4298, 5592, 1328, 9552, 30594, 19334, 16142, 15684, 81464, 28230, 31908, 19610, 35618, 82074, 44294, 43332, 34062, 89690, 162144, 134514, 173360, 31398, 404598, 212702, 188030, 542604, 265622

Offset: 1

Jon E. Schoenfield, Jan 17 2021

nonn

If "integers having at least 4 divisors each" in this sequence's definition were replaced with "integers having at least 3 divisors each" (i.e., composite numbers), the resulting sequence would be A045881.
A045881(n) = a(n) except when the run of 2*n-1 consecutive composite numbers beginning with A045881(n) includes a number with exactly 3 divisors (i.e., the square of a prime). The first six such exceptions are as follows:
.
         n  A045881(n)  a(n)  3-divisor number
        --  ----------  ----  ----------------
         1        4        6        4 =  2^2
         2        8       14        9 =  3^2
         3       24       32       25 =  5^2
         7      114      294      121 = 11^2
         9      524     1070      529 = 23^2
        12     1670     2180     1681 = 41^2
.
There are no other exceptions among the first 672 terms of A045881 (see the b-file there). Can it be proved that there are no other exceptions?

			a(1)=6 because 6=2*3 (which has 4 divisors, {1,2,3,6}) is the first isolated number that has at least 4 divisors.
a(2)=14 because 14 is the first number that begins a run of exactly 2*2-1=3 consecutive integers having at least 4 divisors each: tau(14)=tau(2*7)=4; tau(15)=tau(3*5)=4; tau(16)=tau(2^4)=5.
a(3)=32 because 32 is the first number that begins a run of exactly 2*3-1=5 consecutive integers having at least 4 divisors each: tau(32)=tau(2^5)=6; tau(33)=tau(3*11)=4; tau(34)=tau(2*17)=4; tau(35)=tau(5*7)=4; tau(36)=tau(2^2*3^2)=9.

Cf. A045881.

A271233 Composite integers sandwiched between primes p, q with q-p = 8.

Original entry on oeis.org

90, 91, 92, 93, 94, 95, 96, 360, 361, 362, 363, 364, 365, 366, 390, 391, 392, 393, 394, 395, 396, 402, 403, 404, 405, 406, 407, 408, 450, 451, 452, 453, 454, 455, 456, 480, 481, 482, 483, 484, 485, 486, 492, 493, 494, 495, 496, 497, 498, 684, 685, 686, 687, 688, 689, 690

Offset: 1

Michel Marcus, Apr 02 2016

nonn

			The composite number 90 is sandwiched between consecutive primes 89 and 97, and 97-89=8, so 90 is a member of the sequence.

Laurent Coppey, Décompositions multiplicatives directes des entiers, Diagrammes, 65-66 (2011), p. 1-68, in French, see J8 p. 11.

Cf. A014574, A031926, A045881, A271211, A271232.

Range[#[[1]]+1,#[[2]]-1]&/@Select[Partition[Prime[Range[150]],2,1],#[[2]]-#[[1]] == 8&]//Flatten (* Harvey P. Dale, May 15 2022 *)

lista(nn) = {forcomposite(c=4, nn, if ((p=precprime(c)) && ((nextprime(c)-p)==8), print1(c, ", ")););}

A343729 a(n) is the smallest number that starts a run of exactly n consecutive integers that are neither primes nor semiprimes.

Original entry on oeis.org

8, 27, 98, 342, 242, 6092, 3174, 2870, 2522, 16023, 26604, 114359, 41780, 74492, 39344, 463162, 104660, 248408, 517422, 923723, 506510, 1930824, 584214, 2560178, 4036968, 4570412, 4552364, 7879254, 4417814, 27841052, 5167588, 13683035, 9725108, 47735343

Offset: 1

Jon E. Schoenfield, May 22 2021

nonn

			a(5) = 242 because 241 is a prime, 247=13*19 is a semiprime, none of the 5 consecutive integers between them are primes or semiprimes (242 = 2*11^2, 243 = 3^5, 244 = 2^2*61, 245 = 5*7^2, and 246 = 2*3*41), and there is no smaller run of exactly 5 consecutive integers that are neither prime nor semiprime.

Cf. A045881, A340735.

Module[{nn=48*10^6,lst},lst=Table[Which[PrimeQ[n],1,PrimeOmega[n]==2,1,True,0],{n,nn}]; Flatten[Table[SequencePosition[lst,Join[{1},PadRight[{},n,0],{1}],1],{n,34}],1]][[;;,1]]+1 (* Harvey P. Dale, May 30 2024 *)

Corrected by Harvey P. Dale, May 30 2024

cp's OEIS Frontend

A000230 a(0)=2; for n>=1, a(n) = smallest prime p such that there is a gap of exactly 2n between p and next prime, or -1 if no such prime exists.

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A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1.

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A271211 Composite integers sandwiched between primes p, q with q-p = 4.

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A075067 Start of the first run of exactly n consecutive odd composite numbers.

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A271232 Composite integers sandwiched between primes p, q with q-p = 6.

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A340735 a(n) is the smallest positive integer that begins a run of exactly 2*n-1 consecutive integers having at least 4 divisors each.

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A271233 Composite integers sandwiched between primes p, q with q-p = 8.

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A343729 a(n) is the smallest number that starts a run of exactly n consecutive integers that are neither primes nor semiprimes.

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