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.

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

Links

Table of n, a(n) for n=1..52.

Michael M. Ross The High Primality of Prime-Derived Quadratic Sequences (2007)

Michael M. Ross How to Use Qtest (2007)

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

Cf. A126665, A126719.

Sequence in context: A191041 A106932 A007635 * A205700 A228070 A289685

Adjacent sequences: A140944 A140945 A140946 * A140948 A140949 A140950

Keyword

nonn,uned,tabf

Author

Michael M. Ross, Jul 24 2008

Status

approved