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A323740
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a(n) is the smallest prime p such that p + 6*t is also prime for every triangular number t up to, but not including, the n-th triangular number (or 0 if no such prime exists).
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0
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2, 7, 13, 5, 53, 1033, 821, 10133, 9461, 11, 105276481, 2201568973, 35401, 10445256498283, 1945187598443, 35849093549903
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OFFSET
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1,1
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COMMENTS
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For n > 16, a(22)=23 is the only nonzero term up to 10^14. - Bert Dobbelaere, Apr 08 2019
An arithmetic progression such as {m, m+k, m+2k, m+3k, ...} where q is the largest prime that does not divide k cannot have more than q-1 successive prime terms (unless m = q). But is there a limit to the number of successive terms of a quadratic sequence like {m, m+k, m+3k, m+6k, m+10k, ..., m+(j*(j+1)/2)*k, ...}, starting with the initial term m, that can be prime?
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LINKS
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EXAMPLE
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a(1)=2 because 2 is prime but 2 + 6*T(1) = 2 + 6*1 = 8 is not prime, and there is no smaller prime for which this is the case.
a(4)=5 because 5, 5 + 6*1 = 11, 5 + 6*3 = 23, and 5 + 6*6 = 41 are all prime, but 5 + 6*10 = 65 is not prime, and there is no smaller prime for which this is the case.
a(22)=23: 23, 29, 41, 59, 83, 113, 149, 191, 239, 293, 353, 419, 491, 569, 653, 743, 839, 941, 1049, 1163, 1283, and 1409 are all prime, but 1541 is not.
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PROG
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(PARI) isok(p, n) = {my(t=0); for (k=1, n-1, t += k; if (! isprime(p+6*t), return (0)); ); t += n; !isprime(p+6*t); }
a(n) = {my(p=2); while (! isok(p, n), p = nextprime(p+1)); p; } \\ Michel Marcus, Mar 11 2019
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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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EXTENSIONS
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STATUS
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approved
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