A014320 -id:A014320 - cp's OEIS frontend

A335366 Primes preceding the first-occurrence gaps in A014320.

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

Offset: 1

Alexei Kourbatov, Jun 03 2020

nonn

Contains A002386 as a subsequence. First differs from A002386 at a(7)=139. This sequence is a permutation of all positive terms of A000230, in increasing order. See 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)=2. The next prime is 5, and the next gap is 5 - 3 = 2; this gap size has not occurred before, so a(2)=3. 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. A000230, A002386, A014320, A330853, A330854, A334543, A334544, A335367.

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

a(n) = A335367(n) - A014320(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).

A131702 Distances between the locations of new prime gaps (A014320).

Original entry on oeis.org

0, 1, 4, 14, 5, 3, 11, 52, 54, 34, 27, 45, 18, 84, 61, 160, 147, 444, 647, 47, 311, 33, 851, 224, 82, 41, 216, 148, 728, 89, 3357, 57, 659, 3853, 1814, 504, 920, 1222, 2019, 4256

Offset: 1

Giovanni Teofilatto, Sep 16 2007

nonn

A014320 lists "new" gaps in the sequence A001223 of prime gaps (not necessarily records as A005669 does).
The locations of these new gaps in A001223 are 1, 2, 4, 9, 24, 30, 34,...
The present sequence lists the first difference of these locations, minus 1: a(1) = 2-1-1. a(2)=4-2-1. a(3)=9-4-1. a(4)=24-9-1.
The sequence therefore argues: need to skip 0 in A001223 to reach a new gap, need to skip 1 to reach a new gap, need to skip 4 to reach a new gap...

Cf. A001223.

A001223 := proc(n) option remember; ithprime(n+1)-ithprime(n) ; end proc:
A014320 := proc(n) option remember; if n = 1 then return 1; else for k from 1 do t := A001223(k) ; isn := true; for i from 1 to n-1 do if procname(i) = t then isn := false; end if; end do: if isn then return t; end if; end do: end if; end proc:
locng := proc(n) option remember; g := A014320(n) ; for k from 1 do if A001223(k) = g then return k; end if; end do: end proc:
A131702 := proc(n) locng(n+1)-locng(n)-1 ; end proc: seq(A131702(n),n=1..40) ;

More terms, program and comment by R. J. Mathar, Aug 23 2010

A187779 Numbers n for which the terms of A014320 up to n=A014320(k) are a permutation of {1,2,4,6,...,2*(k-1)}.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 32, 36, 150

Offset: 1

Peter J. C. Moses, Jan 05 2013

nonn

An old conjecture states that the equation prime(n+1)-prime(n)=2m is solvable for m=1/2,1,2,... . a(n) indicate the smallest term of A014320 such that among the previous terms there occur 1 and all previous even numbers<=a(n). The primes up to 10^8 do not generate any further terms.

According to Thomas R. Nicely (see Links) the next term, if it exists, must be greater than 1344 and equal to a gap occurring between primes greater than 4*10^18. - Giovanni Resta, Jan 06 2013

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

Essentially equivalent to A179985.

Definition corrected by Giovanni Resta, Jan 05 2013

A127480 A131702 and A014320 interleaved.

Original entry on oeis.org

0, 2, 1, 4, 4, 6, 14, 8, 5, 14, 3, 10, 11, 12, 52, 18, 54, 20, 34, 22, 27, 34, 45, 24, 18, 16, 84, 26, 61, 28, 160, 30, 147, 32, 444, 36, 647, 44, 47, 42, 311, 40, 33, 52, 851, 48, 224, 38, 82, 72, 41, 50, 216, 62, 148, 54, 728, 60, 89, 58

Offset: 1

Giovanni Teofilatto, Sep 12 2007

a(2n)= A014320(n+1). a(2n-1) = A131702(n).

Formulas added, sequence extended, arbitrarily defined first term replaced - R. J. Mathar, Aug 23 2010

A330853 First occurrences of gaps between primes 6k+1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 54, 42, 36, 48, 60, 78, 66, 72, 84, 90, 96, 114, 102, 162, 108, 126, 120, 132, 150, 138, 144, 174, 168, 156, 192, 204, 180, 198, 252, 270, 216, 222, 186, 228, 210, 240, 282, 246, 234, 276, 264, 258, 312, 330, 318, 288, 306, 294, 336, 300, 378

Offset: 1

Alexei Kourbatov, Apr 27 2020

nonn

Contains A268925 as a subsequence.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

			The first primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. The next prime 6k+1 is 19, and the gap 19-13=6 already occurred, so a new term is not added to the sequence. The next prime 6k+1 is 31, and the gap 31-19=12 is occurring for the first time; therefore a(2)=12.

Alexei Kourbatov, Table of n, a(n) for n = 1..135
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.

Cf. A002476, A014320, A058320, A330854 (primes 6k+1 preceding the first-occurrence gaps), A330855 (primes 6k+1 at the end of the first-occurrence gaps).

isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

a(n) = A330855(n) - A330854(n).

A058320 Distinct even prime-gap lengths (number of composites between primes), from 3+2, 7+4, 23+6,...

Original entry on oeis.org

2, 4, 6, 8, 14, 10, 12, 18, 20, 22, 34, 24, 16, 26, 28, 30, 32, 36, 44, 42, 40, 52, 48, 38, 72, 50, 62, 54, 60, 58, 46, 56, 64, 68, 86, 66, 70, 78, 76, 82, 96, 112, 100, 74, 90, 84, 114, 80, 88, 98, 92, 106, 94, 118, 132, 104, 102, 110, 126, 120, 148, 108

Offset: 0

Warren D. Smith, Dec 11 2000

Nicely and Nyman have sieved up to 1.3565*10^16 at least. They admit it is likely they have suffered from hardware or software bugs, but believe the probability the sequence up to this point is incorrect is <1 in a million. This sequence is presumably all even integers (in different order). It is not monotonic. The monotonic subsequence of record-breaking prime gaps is A005250.

Essentially the same as A014320. [From R. J. Mathar, Oct 13 2008]

Richard P. Brent, The first occurrence of large gaps between successive primes, Math. Comp. 27:124 (1973), 959-963.
Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Cf. A008996, A005250.

Equals 2*A014321(n-1).

DeleteDuplicates[Differences[Prime[Range[2,200000]]]] (* Harvey P. Dale, Dec 07 2014 *)

Comment corrected by Harvey P. Dale, Dec 07 2014

A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 36, 42, 54, 48, 60, 84, 66, 78, 72, 126, 90, 102, 108, 114, 96, 120, 150, 138, 162, 132, 144, 168, 246, 156, 180, 186, 240, 204, 192, 216, 198, 210, 174, 258, 252, 222, 234, 228, 318, 282, 264, 276, 342, 306, 294, 312, 270, 354, 372

Offset: 1

Alexei Kourbatov, May 05 2020

nonn

Contains A268928 as a subsequence. First differs from A268928 at a(5)=24.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 has not occurred before, so a(2)=12.

Alexei Kourbatov, Table of n, a(n) for n = 1..160
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.

Cf. A007528, A014320, A058320, A268928, A330853, A334544, A334545.

isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

a(n) = A334545(n) - A334544(n).

A330854 Primes of the form 6k + 1 preceding the first-occurrence gaps in A330853.

Original entry on oeis.org

7, 19, 43, 241, 283, 1327, 1489, 1951, 2389, 4363, 7789, 10177, 16759, 22189, 24247, 38461, 40237, 43441, 69499, 75403, 100801, 118927, 171271, 195541, 204163, 250279, 480169, 577639, 590437, 1164607, 1207699, 1278817, 1382221, 1467937, 1526659, 1889803, 2314369

Offset: 1

Alexei Kourbatov, Apr 27 2020

nonn

Subsequence of A002476. First differs from A268926 in that that sequence does not include 283; all terms of A268926 are in this sequence but many terms of this sequence are not in A268926.

			The first two primes of the form 6k + 1 are 7 and 13, so a(1) = 7. The next prime of that form is 19, and the gap 19 - 13 = 6 already occurred; so a new term is not added to the sequence. The next prime of the form 6k + 1 is 31, and the gap 31 - 19 = 12 is occurring for the first time; therefore a(2) = 19.
The gap between 241 and the next prime of the form 6k + 1 (271) is 30. So 241 is in the sequence.
Although the gap between 283 and 307 is only 24 (which is less than 30), the gap is of a size not previously encountered. So 283 is in the sequence.

Alexei Kourbatov, Table of n, a(n) for n = 1..135
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.

Cf. A002476, A014320, A058320, A330853 (gap sizes), A330855.

isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(s", "); isFirstOcc[g/6]=0); s=p)

a(n) = A330855(n) - A330853(n).

A334544 Primes of the form 6k - 1 preceding the first-occurrence gaps in A334543.

Original entry on oeis.org

5, 29, 113, 197, 359, 521, 1109, 1733, 4289, 6389, 7349, 8297, 9059, 12821, 35603, 37691, 58787, 59771, 97673, 105767, 130649, 148517, 153749, 180797, 220019, 328127, 402593, 406907, 416693, 542261, 780401, 1138127, 1294367, 1444271, 1463621, 1604753

Offset: 1

Alexei Kourbatov, May 05 2020

nonn

Subsequence of A007528. Contains A268929 as a subsequence. First differs from A268929 at a(5)=359.

A334543 lists the corresponding gap sizes; see more comments there.

			The first two primes of the form 6k-1 are 5 and 11; we have a(1)=5. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap size 41-29=12 has not occurred before, so a(2)=29.

Alexei Kourbatov, Table of n, a(n) for n = 1..160
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.

Cf. A007528, A014320, A058320, A268929, A330854, A334543, A334545.

isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(s", "); isFirstOcc[g/6]=0); s=p)

a(n) = A334545(n) - A334543(n).

cp's OEIS Frontend

A335366 Primes preceding the first-occurrence gaps in A014320.

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

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A131702 Distances between the locations of new prime gaps (A014320).

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A187779 Numbers n for which the terms of A014320 up to n=A014320(k) are a permutation of {1,2,4,6,...,2*(k-1)}.

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A127480 A131702 and A014320 interleaved.

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A330853 First occurrences of gaps between primes 6k+1: gap sizes.

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A058320 Distinct even prime-gap lengths (number of composites between primes), from 3+2, 7+4, 23+6,...

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Mathematica

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A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

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A330854 Primes of the form 6k + 1 preceding the first-occurrence gaps in A330853.

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