A114200 When the n-th term of this sequence is added to or subtracted from the square of the n-th prime of the form 4k + 1 (i.e., A002144(n)), the result in both cases is a square.

24, 120, 240, 840, 840, 720, 2520, 1320, 5280, 6240, 9360, 3960, 10920, 3360, 18480, 14280, 22440, 17160, 6840, 31920, 10920, 26520, 43680, 50160, 16320, 35880, 57960, 73920, 38760, 15600, 46200, 100800, 107640, 122400, 138600, 128520, 148200

Offset

1,1

Comments

This sequence and A002144 give rise to a class of monic polynomials x^2 + bx + c where b = +- A002144(n) and c = +- a(n)/4 that will factor over the integers regardless of the sign of c. For example, x^2 - 13x - 30 and x^2 - 13x + 30 are two such polynomials. Further polynomials with this property can be found by transforming the roots.

Links

Table of n, a(n) for n=1..37.

Example

a(2) = 120 and A002144(2) = 13. 13^2 - 120 = 7^2 and 13^2 + 120 = 17^2.

Prog

(PARI) getpr(n) = {nb = 0; p = 2; while (nb != n, p = nextprime(p+1); if ((p % 4) == 1, nb++); ); p; }
a(n) = {p = getpr(n); psq = p^2; k = 1; while (!issquare(psq+k) || !issquare(psq-k), if (k>psq, k = 0; break); k++; ); k; } \\ Michel Marcus, Sep 25 2013

Crossrefs

Cf. A002144.

Sequence in context: A369541 A198438 A256629 * A229567 A069074 A360389

Adjacent sequences: A114197 A114198 A114199 * A114201 A114202 A114203

Keyword

nonn

Author

Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005

Extensions

Definition corrected by Zak Seidov, Jul 20 2010

a(17) corrected by Zachary Sizer, Jan 01 2025

Status

approved