A005250 -id:A005250 - cp's OEIS frontend

A350095 a(n) is the smaller of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

13, 31, 89, 199, 211, 887, 1129, 1327, 9973, 15683, 19609, 44293, 155921, 370261, 396733, 492113, 1357201, 1671781, 3826019, 17836409, 20831323, 47465267, 107534587, 122164747, 434865437, 436273009, 2300942549, 4302407359, 10726904659, 25056082087, 42652618343

Offset: 1

Hugo Pfoertner, Dec 25 2021

nonn

			a(1) = 13: semiprime 15 < 17 = nextprime(a(1)) = A350096(1);
a(2) = 31: semiprimes 33, 35 < 37 = A350096(2);
a(6) = 887: semiprimes 889, 893, 895, 899, 901, 905 < 907 = A350096(6);
a(7) = 1129: semiprimes 1133, 1135, 1137, 1139, 1141, 1145, 1147, 1149 < 1151 = A350096(7);
a(8) = 1327: semiprimes 1329, 1333, 1337, 1339, 1343, 1345, 1347, 1349, 1351, 1355, 1357 < 1361 = A350096(8).

Lucas A. Brown, Table of n, a(n) for n = 1..32 (terms 1-28 from _Hugo Pfoertner_, 29-31 from _Martin Ehrenstein_)
Lucas A. Brown, Python program.

A350096 are the upper ends of the intervals, A350097 are the corresponding counts of odd squarefree semiprimes in the intervals.

Cf. A000101, A005250, A046388.

A350096(n) = nextprime(a(n)).

a(29)-a(31) from Martin Ehrenstein, Dec 28 2021

a(32) from Lucas A. Brown, Mar 21 2024

A350096 a(n) is the larger of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

Original entry on oeis.org

17, 37, 97, 211, 223, 907, 1151, 1361, 10007, 15727, 19661, 44351, 156007, 370373, 396833, 492227, 1357333, 1671907, 3826157, 17836561, 20831533, 47465443, 107534789, 122164969, 434865671, 436273291, 2300942869, 4302407713, 10726905041, 25056082543, 42652618807

Offset: 1

Hugo Pfoertner, Dec 25 2021

nonn

			See A350095.

Lucas A. Brown, Table of n, a(n) for n = 1..32 (terms 1-28 from _Hugo Pfoertner_, 29-31 from _Martin Ehrenstein_)
Lucas A. Brown, Python program.

A350097 gives the corresponding counts.

Cf. A000101, A002386, A005250, A046388, A349995, A350095.

a(n) = nextprime(A350095(n)).

a(29)-a(31) from Martin Ehrenstein, Dec 28 2021

a(32) from Lucas A. Brown, Mar 21 2024

A354604 Midpoints of record gaps between primes: a(n) = (A000101(n) + A002386(n))/2 for n > 1.

Original entry on oeis.org

4, 9, 26, 93, 120, 532, 897, 1140, 1344, 9569, 15705, 19635, 31433, 155964, 360701, 370317, 492170, 1349592, 1357267, 2010807, 4652430, 17051797, 20831428, 47326803, 122164858, 189695776, 191912907, 387096258, 436273150, 1294268635, 1453168287, 2300942709, 3842610941, 4302407536, 10726904850, 20678048489, 22367085156, 25056082315, 42652618575

Offset: 2

Donghwi Park, Jul 08 2022

nonn

In the displayed portion of the sequence, the only numbers of least prime signature (A025487) are 4 and 120. This is noteworthy because numbers of least prime signature frequently are adjacent to primes (see A344385). It appears to be far more rare for a number of least prime signature to be at the center of a maximal prime gap. With 4 being a term in A344385, 120 seems to have a unique status. - Hal M. Switkay, Mar 13 2025

Hugo Pfoertner, Table of n, a(n) for n = 2..83
Hugo Pfoertner, Gap sizes vs location, using Plot 2.

Subsequence of A024675.

Cf. A000040, A000101, A002386, A344385.

A038343 Maximal value of difference between successive primes among the first 10^n primes.

Original entry on oeis.org

6, 18, 34, 72, 114, 154, 222, 292, 394, 486, 652, 766

Offset: 1

Enoch Haga

			Among the first 10 primes, {2,3,...,23,29}, the largest difference is 29-23=6. Therefore 6 is the largest prime gap in the first ten primes.

Enoch Haga, Exploring Primes on Your PC, 2nd edition, 1998, ISBN 1-885794-16-9. Table 2, page 33.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].

Cf. A005250.

a = 1; b = 1; d = 0; k = 1; Do[ While[k <= 10^n, a = b; b = Prime[k]; If[b - a > d; d = b - a]; k++ ]; Print[d], {n, 12}] (* Robert G. Wilson v, Sep 24 2004 *)

Edited and extended by Robert G. Wilson v, Sep 24 2004

A100180 Gaps associated with the record prime gaps described in A133429, A133430.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 26, 28, 30, 32, 36, 38, 46, 56, 64, 66, 70, 74, 80, 88, 92, 94, 102, 108, 116, 124, 134, 140, 142, 144, 150, 156, 158, 166, 186, 194, 200, 224, 226, 228, 254, 256, 264, 278, 294, 298, 314, 316, 328, 334, 362, 368, 370, 388

Offset: 1

N. J. A. Sloane, Dec 16 2007

nonn

Prime gaps whose first occurrence is later than all smaller gaps. - Brian Kehrig, Feb 09 2025

N. J. A. Sloane, Table of n, a(n) for n=1..124 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Gaps between consecutive primes
Index entries for primes, gaps between

Cf. A002386, A005250, A000230, A133429, A133430.

A127441 Numbers n such that between n and n+sqrt(n) there are no primes.

Original entry on oeis.org

3, 7, 8, 13, 23, 24, 31, 113, 114, 115, 116

Offset: 1

Artur Jasinski, Jan 14 2007

If Oppermann's conjecture is true, then all terms are nonsquares. Data from A002386 and A005250 show that a(12) > 6787988999657777797 if it exists. Most likely there are no further terms. - Chai Wah Wu, Mar 08 2019

Wikipedia, Oppermann's conjecture

Cf. A000720, A002386, A005250, A028392.

a = {}; Do[If[PrimePi[x + x^(1/2)] - PrimePi[x] == 0, AppendTo[a, x]], {x, 1, 249900000000000}]; a

isok(n) = primepi(n+sqrtint(n)) == primepi(n); \\ Michel Marcus, Nov 07 2013

A338577 Primes p such that A013632(p)*A105161(p) > p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 31, 47, 83, 89, 113, 199, 1327

Offset: 1

Robert Israel, Nov 03 2020

Primes p such that (q-p)*(r-p) > p, where q and r are the next two primes after p.
a(16) > 10^8 if it exists.
Sequence is finite if Cramér's conjecture is true. - Chai Wah Wu, Nov 03 2020
Data from A002386 and A005250 show that a(16) > 18361375334787046697 if it exists. - Jason Yuen, Jun 13 2024

			a(5) = 11 is a member because 11 is prime, the next two primes are 13 and 17, and (13-11)*(17-11) = 12 > 11.

Contains A338567.
Cf. A013632, A105161, A338578.
Cf. A002386, A005250.

p:= 0: q:=2:r:= 3: R:= NULL:
while p < 10^4 do
  p:= q: q:= r: r:= nextprime(r);
  if (q-p)*(r-p) > p then R:= R, p fi
od:
R;

from sympy import nextprime
A338577_list, p, q, r = [], 2,3,5
while p < 10**6:
    if (q-p)*(r-p) > p:
        A338577_list.append(p)
    p, q, r = q, r, nextprime(r) # Chai Wah Wu, Nov 03 2020

A063096 Non-record differences among consecutive primes.

Original entry on oeis.org

10, 12, 16, 24, 26, 28, 30, 32, 38, 40, 42, 46, 48, 50, 54, 56, 58, 60, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108, 110, 116, 120, 122, 124, 126, 128, 130, 134, 136, 138, 140, 142, 144, 146, 150, 152, 156, 158, 160, 162

Offset: 1

Labos Elemer, Aug 07 2001

nonn

These values do not arise in A005250 nor in A063095.

Almost certainly this sequence is exactly the even numbers not in A005250. - Franklin T. Adams-Watters, Oct 09 2006

			10 and 12 are here because after the first gap of 8 (89 to 97), the next larger gap is 14 (113 to 127); thus 10 and 12 are never the largest gap. 11 is not here because it is never the gap between consecutive primes.

Cf. A005250, A001223, A063095.

{ default(primelimit, 4294965247); n=0; r=0; for (m=1, 10^9, g=prime(m + 1) - prime(m); if (g > r, a=r + 2; r=g; while (a < r, write("b063096.txt", n++, " ", a); a+=2); if (n==100, break)) ) } \\ Harry J. Smith, Aug 18 2009

A085500 Indices of primes where nondecreasing gaps occur.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 15, 16, 18, 21, 23, 24, 30, 62, 66, 99, 154, 189, 217, 1059, 1183, 1532, 1663, 1831, 2225, 2810, 3385, 14357, 29040, 30802, 31545, 40933, 103520, 104071, 118505, 149689, 325852, 733588, 983015, 1094421, 1319945, 2850174, 6957876, 10539432, 10655462

Offset: 1

Farideh Firoozbakht, Aug 15 2003

nonn

A005669 is a subsequence of this sequence.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A8, pp. 31-39.

Jeff Young and Aaron Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Cf. A085237, A005669, A005250, A134266.

See also A214935(n) = A000720(A205827(n)), A241540(n) = A000720(A182514(n)).

f[n_] := Prime[n+1]-Prime[n]; v1={}; v2={}; Do[If[f[n]>=If[n==1, 1, Last[v2]], v=n; v1=Append[v1, n]; v2=Append[v2, f[v]]; Print[v1]], {n, 105000000}]

a(n) = A000720(A134266(n)). - M. F. Hasler, Apr 26 2014

a(45)-a(47) from Amiram Eldar, Sep 05 2024

A115401 Record differences between prime(n+3) and prime(n). Records in A031165.

Original entry on oeis.org

5, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 40, 42, 46, 50, 54, 58, 60, 62, 64, 68, 78, 84, 112, 116, 118, 120, 126, 128, 142, 152, 170, 178, 184, 192, 194, 198, 208, 210, 216, 220, 222, 252, 258, 270, 300, 318, 336, 348, 354, 370, 408

Offset: 1

Jonathan Vos Post, Jan 22 2006

This is the k=3 case of the set of sequences "records in a(k,n) = prime(n+k) - prime(n)." The k=1 case is given by A005250 (ncreasing gaps between primes), A000101 [increasing gaps between primes (upper end)] and A002386, which gives lower ends of these gaps. The k=2 case is A031132. The merits of these records are (prime(n+3)-prime(n))/log (prime(n)). The first record merit is 5/log 2 = 16.6096405. The second record merit is 8/log 3 = 16.7672262.

			a(1) = A031165(1) = prime(4) - prime(1) = 7 - 2 = 5, which is the only odd element of this sequence.
a(2) = A031165(2) = prime(5) - prime(2) = 11 - 3 = 8.
a(3) = A031165(4) = prime(7) - prime(4) = 17 - 7 = 10.
a(4) = A031165(7) = prime(10) - prime(7) = 29 - 17 = 12.
a(5) = A031165(9) = prime(12) - prime(9) = 37 - 23 = 14.

Cf. A000040, A000101, A001223, A002386, A005250, A007530, A031131, A031132, A031165.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; d = 0; p = 1; q = 2; r = 3; s = 5; lst = {}; Do[{p, q, r, s} = {q, r, s, NextPrim[s]}; If[s > d + p, d = s - p; AppendTo[lst, d]; Print[d]], {n, 10^8}] (* Robert G. Wilson v *)

Corrected and extended by Robert G. Wilson v, Jan 23 2006

cp's OEIS Frontend

A350095 a(n) is the smaller of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

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A350096 a(n) is the larger of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

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A354604 Midpoints of record gaps between primes: a(n) = (A000101(n) + A002386(n))/2 for n > 1.

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A038343 Maximal value of difference between successive primes among the first 10^n primes.

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Mathematica

Extensions

A100180 Gaps associated with the record prime gaps described in A133429, A133430.

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A127441 Numbers n such that between n and n+sqrt(n) there are no primes.

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Mathematica

PARI

A338577 Primes p such that A013632(p)*A105161(p) > p.

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Maple

Python

A063096 Non-record differences among consecutive primes.

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PARI

A085500 Indices of primes where nondecreasing gaps occur.

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