1,1

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.

Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.

In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. Keep in mind that 12 is a canonical base for mathematics in general since any prime greater than 3 is of the form 6k+-1, any prime of the form 4k+1 is a sum of squares while any prime of the form 4k+3 is never a sum of squares and lcm(6,4)=12. - Walter A. Kehowski (wkehowski(AT)cox.net), May 31 2008

John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

f[x_, y_]:=7*x^2+4*x*y+52*y^2; lst={}; Do[Do[p=f[x, y]; If[PrimeQ[p], AppendTo[lst, p]], {y, -4!, 3*4!}], {x, -4!, 3*4!}]; Take[Union[lst], 90] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009]

Union[QuadPrimes[7, 4, 52, 10000], QuadPrimes[7, -4, 52, 10000]] (* see A106856 *)

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.

Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.

Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.

Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.

Cf. A107007, A107003, A139830, A107169, A139831, A107154, A139847.

Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.

Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.

Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Sequence in context: A165878 A142358 A020477 this_sequence A142400 A032460 A101746

Adjacent sequences: A140630 A140631 A140632 this_sequence A140634 A140635 A140636

nonn,easy

T. D. Noe (noe(AT)sspectra.com), May 19 2008