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
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.
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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)
A268925
Record (maximal) gaps between primes of the form 6k + 1.
Original entry on oeis.org
6, 12, 18, 30, 54, 60, 78, 84, 90, 96, 114, 162, 174, 192, 204, 252, 270, 282, 312, 330, 336, 378, 462, 486, 522, 528, 534, 600, 606, 612, 642, 666, 780, 810, 894, 1002
Offset: 1
The first two primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. The next prime of this form is 19; the gap 19-13 is not a record so nothing is added to the sequence. The next prime of this form is 31; the gap 31-19=12 is a new record, so a(2)=12.
- Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
- Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
- Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
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re = 0; s = 7; Reap[For[p = 13, p < 10^8, p = NextPrime[p], If[Mod[p, 6] != 1, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* Jean-François Alcover, Dec 12 2018, from PARI *)
records[n_]:=Module[{ri=n,m=0,rcs={},len},len=Length[ri];While[len>0,If[ First[ri]>m,m=First[ri];AppendTo[rcs,m]];ri=Rest[ri];len--]; rcs]; records[ Differences[Select[6*Range[0,3*10^6]+1,PrimeQ]]] (* the program generates the first 30 terms of the sequence. *) (* Harvey P. Dale, Dec 19 2021 *)
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re=0; s=7; forprime(p=13, 1e8, if(p%6!=1, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
A268927
Primes 6k + 1 at the end of the maximal gaps in A268925.
Original entry on oeis.org
13, 31, 61, 271, 1381, 4423, 7867, 22273, 24337, 38557, 40351, 69661, 480343, 1164799, 1207903, 1468189, 1526929, 3976003, 11962963, 14466967, 19097593, 30098239, 39895771, 198389797, 303644749, 393202651, 485949787, 680676709, 1917215533, 3868901233, 4899890383, 6957510319, 7599383353
Offset: 1
The first two primes of the form 6k+1 are 7 and 13, so a(1)=13. The next prime of this form is 19; the gap 19-13 is not a record so nothing is added to the sequence. The next prime of this form is 31 and the gap 31-19=12 is a new record, so a(2)=31.
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re=0; s=7; forprime(p=13, 1e8, if(p%6!=1, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)
Showing 1-3 of 3 results.
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