A141167 Primes of the form 8*x^2+x*y-8*y^2.
61, 67, 113, 157, 193, 197, 227, 241, 257, 419, 499, 587, 631, 643, 653, 739, 821, 823, 859, 863, 907, 929, 947, 971, 997, 1019, 1039, 1051, 1087, 1181, 1187, 1217, 1289, 1303, 1319, 1373, 1511, 1531, 1637, 1777, 1783, 1801, 1913, 1997, 2027, 2039, 2069, 2087, 2129, 2213
Offset: 1
Keywords
Examples
a(6)=197 because we can write 197 = 8*5^2+5*1-8*1^2.
References
- Z. I. Borevich and I. R. Shafarevich, Number Theory
Links
- Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
- Peter Luschny, Binary Quadratic Forms
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
- D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
Crossrefs
Numbers of the form 8x^2+xy-8y^2 in A243180.
Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141168 (d=257).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Programs
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Mathematica
q := 8*x^2 + x*y - 8*y^2; pmax = 3000; xmax = xmax0 = 50; ymin = ymin0 = -50; ymax = ymax0 = 50; k = 1.3 (* expansion coeff. for maxima *) ; dx = dy = 2; prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, If[xmax == xmax0, xmax, Floor[k*xmax]], dx}, {y, If[ymin == ymin0, ymin, Floor[k*ymin]], If[ymax == ymax0, ymax, Floor[k*ymax]]}, dy]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; Print[Length[prms], " terms", " xmax = ", xmax, " ymin = ", ymin, " ymax = ", ymax ]]; A141167 = prms (* Jean-François Alcover, Oct 26 2016 *) -
Sage
# The function binaryQF is defined in the link 'Binary Quadratic Forms'. Q = binaryQF([8, 1, -8]) print(Q.represented_positives(2213, 'prime')) # Peter Luschny, Oct 26 2016
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