

A140947


Fourcolumned array read by rows: each row gives a series of 4 consecutive primes that share a 2nddegree polynomial relationship and produce a positiveonly 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
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OFFSET

1,1


COMMENTS

These "proximateprime polynomials" exhibit high prime densities. Of the 333 under 100000, 46 have greater than 50% prime values for the first 1000 terms. 2221 positiveonly PPPs have been found under 1000000. All positiveinteger PPPs have complex roots (only negativeinteger 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 PrimeDerived 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



