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Search: id:A140947
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| A140947 |
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Four-columned array read by rows: each row gives a series of 4 consecutive primes that share a 2nd-degree polynomial relationship and produce a positive-only integer series from the derived quadratic. |
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+0
2
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| 17, 19, 23, 29, 41, 43, 47, 53, 79, 83, 89, 97, 227, 229, 233, 239, 347, 349, 353, 359, 349, 353, 359, 367, 379, 383, 389, 397, 439, 443, 449, 457, 569, 571, 577, 587, 641, 643, 647, 653, 673, 677, 683, 691, 677, 683, 691, 701, 1031, 1033, 1039, 1049
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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These "proximate-prime polynomials" exhibit high prime densities. Of the 333 under 100000, 46 have greater than 50% prime values for the first 1000 terms. 2221 positive-only PPPs have been found under 1000000. All positive-integer PPPs have complex roots (only negative-integer PPPs, which are excluded) have real roots. The roots mostly have a real part of 1/2 or a multiple of 1/2.
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REFERENCES
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Purple Math: Finding the Next Number in a Sequence: The Method of Common Differences http://www.purplemath.com/modules/nextnumb.htm
Robert Sacks, Method of Common Differences http://www.numberspiral.com/p/common_diff.html
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LINKS
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Michael M. Ross The High Primality of Prime-Derived Quadratic Sequences (2007)
Michael M. Ross How to Use Qtest (2007)
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FORMULA
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Method of common differences: if (P2 - P1) - (P3 - P2) = (P3 - P2) - (P4 - P3) then polynomial is degree 2.
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EXAMPLE
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For 17, 19, 23, 29 the method of common differences produces coefficients of 1, -1 and 17 for a polynomial expression of n^2 - n + 17.
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CROSSREFS
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Cf. A126665, A126719.
Sequence in context: A106933 A106932 A007635 this_sequence A144487 A108266 A102325
Adjacent sequences: A140944 A140945 A140946 this_sequence A140948 A140949 A140950
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KEYWORD
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nonn,uned,tabf
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AUTHOR
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Michael M. Ross (michaelmross(AT)gmail.com), Jul 24 2008
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