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A269261
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Record (maximal) gaps between primes of the form 10k + 9.
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2
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10, 30, 80, 100, 110, 120, 170, 180, 190, 240, 270, 280, 290, 330, 360, 370, 500, 510, 610, 620, 630, 670, 700, 730, 840, 870, 950, 990, 1020, 1130, 1220, 1280, 1320, 1610, 1770, 1910, 2450
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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 suggests that average gaps between primes of the form 10k + 9 below x are about phi(10)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(10)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(10)=4.
Conjecture: a(n) < phi(10)*log^2(A269263(n)) almost always.
A269262 lists the primes preceding the maximal gaps.
A269263 lists the corresponding primes at the end of the maximal gaps.
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LINKS
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EXAMPLE
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The first two primes of the form 10k + 9 are 19 and 29, so a(1)=29-19=10. The next prime of this form is 59 and the gap 59-29=30 is a new record, so a(2)=30.
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PROG
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(PARI) re=0; s=19; forprime(p=29, 1e8, if(p%10!=9, 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
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AUTHOR
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STATUS
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approved
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