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Search: id:A131626
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| A131626 |
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Prime subsequences in the evaluation of the first derivatives of quadratic equations with positive successive prime coefficients. |
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1
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| 7, 11, 17, 11, 17, 19, 29, 47, 67, 101, 23, 31, 47, 43, 47, 71, 59, 89, 67, 101, 167, 71, 107, 79, 83, 103, 107, 127, 191, 317, 131, 197, 139, 151, 227, 163, 167, 251, 179, 269, 191, 199, 211, 317, 223, 227, 239, 359, 251, 263, 271, 283, 307, 461, 311, 467, 331
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Conjecture: The number of primes in a row for f'(x,p) = 2*prime(p)*x + prime(p+1) is finite.
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FORMULA
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Define prime(p) = the p-th prime number. The equation f(x,p) = prime(p)x^2 + prime(p+1)x + prime(p+2) is differentiated to get f'(x,p) = 2prime(p)x + prime(p+1). Then f'(x,p) is evaluated at p=1,2,.. for each x =1,2,.. until f'(x,p) is not prime at which point x is incremented and p=1,2,..
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EXAMPLE
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For x = 4 we have 2*prime(p)*x + prime(p+1) =
2*2*4+3 = 19 prime,
2*3*4+5 = 29 prime,
2*5*4+7 = 47 prime,
2*7*4+11 = 67 prime,
2*11*4+13 = 101 prime,
2*13*4+17 = 121 not prime.
So the subsequence 19,29,47,67,101 is in the sequence beginning in the 6-th position.
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PROGRAM
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(PARI) g(n) = { local(x, y, p); for(x=0, n, for(p=1, n, y=2*prime(p)*x+prime(p+1); if(isprime(y), print1(y", "), break) ) ) }
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CROSSREFS
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Sequence in context: A089997 A129188 A022950 this_sequence A086762 A076045 A101618
Adjacent sequences: A131623 A131624 A131625 this_sequence A131627 A131628 A131629
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)hotmail.com), Sep 03 2007
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