cp's OEIS Frontend

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.

%I A114200 #15 Jan 01 2025 09:49:57
%S A114200 24,120,240,840,840,720,2520,1320,5280,6240,9360,3960,10920,3360,
%T A114200 18480,14280,22440,17160,6840,31920,10920,26520,43680,50160,16320,
%U A114200 35880,57960,73920,38760,15600,46200,100800,107640,122400,138600,128520,148200
%N 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.
%C A114200 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.
%e A114200 a(2) = 120 and A002144(2) = 13. 13^2 - 120 = 7^2 and 13^2 + 120 = 17^2.
%o A114200 (PARI) getpr(n) = {nb = 0; p = 2; while (nb != n, p = nextprime(p+1); if ((p % 4) == 1, nb++);); p;}
%o A114200 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
%Y A114200 Cf. A002144.
%K A114200 nonn
%O A114200 1,1
%A A114200 Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
%E A114200 Definition corrected by _Zak Seidov_, Jul 20 2010
%E A114200 a(17) corrected by _Zachary Sizer_, Jan 01 2025