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A140947
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.
1
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
OFFSET
1,1
COMMENTS
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.
REFERENCES
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
FORMULA
Method of common differences: if (P2 - P1) - (P3 - P2) = (P3 - P2) - (P4 - P3) then polynomial is degree 2.
EXAMPLE
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.
CROSSREFS
Sequence in context: A191041 A106932 A007635 * A205700 A228070 A289685
KEYWORD
nonn,uned,tabf
AUTHOR
Michael M. Ross, Jul 24 2008
STATUS
approved