login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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 A144487

Adjacent sequences:  A140944 A140945 A140946 * A140948 A140949 A140950

KEYWORD

nonn,uned,tabf

AUTHOR

Michael M. Ross, Jul 24 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 28 06:23 EDT 2017. Contains 287212 sequences.