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A269234
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Record (maximal) gaps between primes of the form 10k + 3.
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2
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10, 20, 50, 70, 80, 90, 100, 110, 120, 130, 150, 300, 360, 420, 500, 510, 540, 550, 610, 630, 650, 690, 780, 810, 820, 840, 870, 890, 960, 990, 1280, 1370, 1380, 1470, 1690
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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 + 3 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(A269236(n)) almost always.
A269235 lists the primes preceding the maximal gaps.
A269236 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 + 3 are 3 and 13, so a(1)=13-3=10. The next prime of this form is 23; the gap 23-13 is not a record so nothing is added to the sequence. The next prime of this form is 43 and the gap 43-23=20 is a new record, so a(2)=20.
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PROG
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(PARI) re=0; s=3; forprime(p=13, 1e8, if(p%10!=3, 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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