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A268925
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Record (maximal) gaps between primes of the form 6k + 1.
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4
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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
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OFFSET
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1,1
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COMMENTS
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Dirichlet's theorem on arithmetic progressions and the GRH suggest that average gaps between primes of the form 6k + 1 below x are about phi(6)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(6)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(6)=2.
Conjecture: a(n) < phi(6)*log^2(A268927(n)) almost always.
Conjecture: phi(6)*n^2/6 < a(n) < phi(6)*n^2 almost always. - Alexei Kourbatov, Nov 27 2019
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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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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PROG
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(PARI) 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)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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