A000101 -id:A000101 - cp's OEIS frontend

A214925 Number of primes <= A214757(n).

5, 5, 5, 10, 25, 31, 104, 159, 190, 219, 1186, 1832, 2227, 3388, 14358, 30804, 31547, 40935, 103522, 104072, 149690, 325853, 1094426, 1319950, 2850175, 6957880, 10539433, 10655464

Offset: 1

John W. Nicholson, Aug 06 2012

			A214757(4) = 29, so a(4) = primepi(A214757(4)) = primepi(29) = 10.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214924, A214926, A214757.

a(n) = pi(A214757(n)) = A000217(A214757(n)).

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A214926 Difference A214925(n) - A214924(n), prime count between Ramanujan primes bounding maximal gap primes.

Original entry on oeis.org

4, 4, 4, 3, 5, 3, 8, 7, 5, 7, 7, 3, 10, 6, 8, 24, 19, 6, 24, 25, 16, 8, 30, 17, 12, 13, 12, 11

Offset: 1

John W. Nicholson, Aug 06 2012

nonn

Conjecture: For every n > 0, a(n) > 1.

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 A001223(n) = p_(n+1) - p_n <= p_rho(m+1) - p_rho(m) = A182873(m). For all rho(m) = A179196(m), A001223(rho(m)) < A165959(m). (Comment copied from A001223). John W. Nicholson, Nov 17 2013

			a(4) = pi(A214757(4)) - pi(A214756(4)) = 10 - 7 = 3

Cf. A214924, A214925, A214756, A214757.

Cf. A000101, A104272, A001223, A002386.

a(n) = pi(A214757(n)) - pi(A214756(n)).

a(n) = rho(A214757(n)) - rho(A214756(n)).

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A086978 Increasing peaks in the prime gap sequence A001632.

Original entry on oeis.org

211, 1847, 5623, 30631, 81509, 82129, 162209, 173429, 404671, 542683, 544367, 1101071, 1444411, 2238931, 5845309, 6752747, 6958801, 11981587, 13626407, 49269739, 83751287, 147684323, 166726561, 378044179, 895858267, 1872852203

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the larger of the two consecutive primes having a late occurring prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime. Also, the next even gap g+2 also occurs earlier.

			1847 is in this list because the previous prime is 1831, giving a prime gap of 16. All even gaps less than 16 occur before this (for smaller primes) and the next even gap, 18, also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A000230, A001223, A001632, A038664, A086977, A086979, A086980, A002386.

A167236 Larger prime power associated with gaps in A121492.

Original entry on oeis.org

2, 7, 16, 23, 37, 59, 97, 149, 211, 307, 907, 1151, 1361, 5623, 8501, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779

Offset: 1

Michael B. Porter, Nov 01 2009, Nov 03 2009

nonn

			59 is in the sequence since 53 and 59 are consecutive prime powers with a difference of 6 and no smaller pair of consecutive prime powers differ by 6 or more.

Jan Kristian Haugland, Table of n, a(n) for n = 1..87. The extra terms after a(33) are copied from A000101 as the associated prime gaps do not contain any prime powers.

Size of gap: A121492
Smaller prime power (start of gap): A002540
Gaps between prime powers: A057820
List of prime powers: A000961

isA000961(n) = (omega(n) == 1 || n == 1)
d_max=0;n_prev=1;for(n=2,1e6,if(isA000961(n),d=n-n_prev;if(d>d_max,print(n);d_max=d);n_prev=n))

a(34) onwards from Jan Kristian Haugland, Oct 18 2024

A185439 Emirp gaps: Differences between consecutive emirps.

Original entry on oeis.org

4, 14, 6, 34, 2, 6, 18, 10, 6, 36, 8, 10, 12, 20, 112, 26, 10, 12, 30, 312, 8, 24, 6, 4, 8, 10, 8, 138, 30, 4, 12, 14, 4, 12, 8, 18, 12, 10, 2, 28, 8, 22, 6, 6, 6, 42, 2, 28, 12, 8, 12, 4, 6, 6, 2, 6, 12, 10, 20, 4, 18, 20, 60, 18, 10, 20, 10, 14, 18, 16, 12, 12, 12, 36, 24, 14, 4, 18, 38, 12, 54, 10, 8, 12, 36, 22, 20

Offset: 1

Jonathan Vos Post, Feb 03 2011

Gaps between consecutive primes whose reversal is a different prime. This is to Differences between consecutive primes (A001223) as emirps (A006567) are to primes (A000040). This was indirectly suggested to me in a facebook conversation with Kevin L. Schwartz. One may use this to derive other sequences: records in emirp gaps; lower of pair of consecutive emirps with record gap; larger of pair of emirps with record gaps, by analogy with A005250, A002386, A000101.

			The first 9 emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107.
Hence the first 8 gaps between consecutive emirps are:
   17 - 13 =  4;
   31 - 17 = 14;
   37 - 31 =  6;
   71 - 37 = 34;
   73 - 71 =  2 (i.e., 71 and 73 are a pair of "twin prime emirps");
   79 - 73 =  6;
   97 - 79 = 18;
  107 - 97 = 10.
So far, we see a minimum gap of 2, and a maximum of 34.

Metin Sariyar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000101, A001223, A002386, A006567.

emirpQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn];idn!=ridn&&PrimeQ[FromDigits[ridn]]]
Take[Differences[Select[Prime[Range[1000]],emirpQ]],90]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = A006567(n+1) - A006567(n).

A335367 Primes at the end of the first-occurrence gaps in A014320.

Original entry on oeis.org

3, 5, 11, 29, 97, 127, 149, 211, 541, 907, 1151, 1361, 1693, 1847, 2503, 2999, 4327, 5623, 9587, 15727, 16183, 19373, 19661, 28277, 30631, 31469, 31957, 34123, 35671, 43391, 44351, 81509, 82129, 89753, 134581, 156007, 162209, 173429, 188107, 212777, 265703

Offset: 1

Alexei Kourbatov, Jun 03 2020

nonn

Contains A000101 as a subsequence. First differs from A000101 at a(7)=149. See A000101, A002386 and A005250 for more references and links.

			The first two primes are 2 and 3, and the first prime gap is 3 - 2 = 1; so a(1)=3. The next prime is 5, and the next gap is 5 - 3 = 2; this gap size has not occurred before, so a(2)=5. The next prime is 7, and the next gap is 7 - 5 = 2; the gap size 2 has already occurred before, so nothing is added to the sequence.

Brian Kehrig, Table of n, a(n) for n = 1..777 (Terms 1..745 from Alexei Kourbatov).
Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.
The Prime Gap List, First Occurrence Prime Gaps

Cf. A000101, A002386, A005250, A014320, A330853, A330854, A334543, A334544, A335366.

my(isFirstOcc=vector(9999, j, 1), s=2); forprime(p=3, 1e8, my(g=p-s); if(isFirstOcc[g], print1(p, ", "); isFirstOcc[g]=0); s=p)

a(n) = A335366(n) + A014320(n).

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

Original entry on oeis.org

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

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

A062530 Smallest prime p such that there is a gap of 2^n between p and previous prime.

Original entry on oeis.org

3, 5, 11, 97, 1847, 5623, 89753, 3851587, 1872852203, 1999066711903, 22790428875365903, 5333419265419188034369535864127397, 55128448018333565337014555712123010955456071077000028555991473847

Offset: 0

Labos Elemer, Jun 25 2001

The next two terms are <= 13615411331526592827872074749865096844383295034548454421 and 768784577114627305753353689789300110953010089817032096740065409732504678169114467301254783622575120297131239844 respectively. - Larry Reeves (larryr(AT)acm.org), Jun 13 2002

			a(2) = 11 because 7 and 11 are consecutive primes with difference 4. - _Sascha Kurz_, Mar 05 2002

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

Cf. A000230, A062529, A151800.

a(n) = {q = 2; p = nextprime(q+1); gap = 2^n; while(p - q != gap, q = p; p = nextprime(p+1)); p;} \\ Michel Marcus, Dec 26 2013

a(n) = A000230(2^(n-1)) + 2^n = Min{p | p-prevprime(p) = 2^n}. - Amarnath Murthy, Feb 24 2002

a(n) = A151800(A062529(n)). - Amiram Eldar, Nov 04 2024

More terms from Sascha Kurz, Mar 05 2002
Further terms from Larry Reeves (larryr(AT)acm.org), Jun 13 2002
Edited by N. J. A. Sloane Aug 31 2009 at the suggestion of R. J. Mathar
a(11)-a(12) calculated from the data at A062529 by Amiram Eldar, Nov 04 2024

cp's OEIS Frontend

A214925 Number of primes <= A214757(n).

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A214926 Difference A214925(n) - A214924(n), prime count between Ramanujan primes bounding maximal gap primes.

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A086978 Increasing peaks in the prime gap sequence A001632.

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A167236 Larger prime power associated with gaps in A121492.

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PARI

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A185439 Emirp gaps: Differences between consecutive emirps.

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Mathematica

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A335367 Primes at the end of the first-occurrence gaps in A014320.

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

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