A084866 -id:A084866 - cp's OEIS frontend

A084865 Primes of the form 2x^2 + 3y^2.

2, 3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 251, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 683, 701, 773, 797, 821, 827, 941, 947, 971, 1013, 1019, 1061, 1091, 1109, 1163, 1181, 1187

Offset: 1

Reinhard Zumkeller, Jun 10 2003

Subsequence of A084864; A084863(a(n))>0.
Conjecture: A084863(a(n))=1?
Is it true that a(n) = A019338(n+1)?
Comment: The truth of the conjecture A084863(a(n))=1 follows from the genus theory of quadratic forms (see Cox, page 61). By comparing enough terms, we see that the conjecture a(n) = A019338(n+1) is false. - T. D. Noe, May 02 2008
Appears to be the primes p such that (p mod 6)*(Fibonacci(p) mod 6)=25. - Gary Detlefs, May 26 2014

			A000040(17) = 59 = 32 + 27 = 2*4^2 + 3*3^2, therefore 59 is a term.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A084863, A084864, A019338, A084866, A139827.

Primes in A002480.

QuadPrimes2[2, 0, 3, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\3), if(isprime(t=w+3*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {2, 3, 5, 11} (mod 24). - T. D. Noe, May 02 2008

A084864 Numbers that can be written in the form 2u^2 + 3v^2, u*v>0.

Original entry on oeis.org

5, 11, 14, 20, 21, 29, 30, 35, 44, 45, 50, 53, 56, 59, 62, 66, 75, 77, 80, 83, 84, 93, 98, 99, 101, 107, 110, 116, 120, 125, 126, 131, 140, 146, 147, 149, 155, 158, 165, 173, 174, 176, 179, 180, 189, 194, 197, 200, 203, 206, 210, 212, 219, 224, 227, 236, 237

Offset: 1

Reinhard Zumkeller, Jun 10 2003

nonn

A084863(a(n)) > 0; for primes see A084865, A084866.

			30 = 18 + 12 = 2*3^2 + 3*2^2, therefore 30 is a term.

Jean-François Alcover, Table of n, a(n) for n = 1..10000

wmax = 240;
Table[w = 2 u^2 + 3 v^2; If[w <= wmax, w, Nothing], {u, 1, Sqrt[wmax/2] // Ceiling}, {v, 1, Sqrt[wmax/3] // Ceiling}] // Flatten // Union (* Jean-François Alcover, Dec 13 2021 *)

A084863 Number of solutions to n = 2u^2 + 3v^2, u*v>0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Jun 10 2003

nonn

a(A084864(n)) > 0.

			n=770 = 2*19^2 + 3*4^2 = 2*17^2 + 3*8^2 = 2*13^2 + 3*12^2 = 2*1 +
3*16^2, therefore a(770)=4.

Eric Weisstein's World of Mathematics, Diophantine Equation: 2nd Powers.

Cf. A084865, A084866.

A259142 Least prime p of the form nq^2+(n+1)r^2 with q and r prime.

Original entry on oeis.org

17, 83, 43, 61, 149, 199, 263, 113, 331, 139, 383, 373, 173, 191, 199, 569, 587, 547, 251, 269, 277, 757, 1223, 1321, 859, 347, 787, 373, 3779, 1789, 1063, 953, 433, 1181, 1019, 1069, 1283, 503, 2311, 5209, 1193, 1453, 563, 1301, 2389, 607, 1367, 1657, 641, 659, 1483, 1777, 1811, 1861, 719, 1913, 1657, 1997, 4391, 3229, 797, 1823

Offset: 1

Zak Seidov, Jun 19 2015

Values of {p,q,r}: {17,3,2},{83,2,5},{43,3,2},{61,2,3},{149,5,2},{199,2,5},{263,3,5},{113,2,3}.
a(2) = A084866(1). - Michel Marcus, Jun 20 2015
For p in A093191, a((p-4)/13) = p. - Robert Israel, Apr 30 2018

			17=1*3^2+2*2^2, 83=2*2^2+3*5^2, 43=3*3^2+4*2^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Zak Seidov, First 100 values of {p,q,r}.

Cf. A000040, A093191.

P:= [seq(ithprime(i),i=1..20)]: np:= 20:
for n from 1 to 100 do
  found:= false;
  while not found do
    R:= sort([seq(seq(n*q^2+(n+1)*p^2,p=P),q=P)]);
    w:= n*4+(n+1)*P[-1]^2+1;
    r:= ListTools:-SelectFirst(isprime,R);
    if r <> NULL and r <= w then
      A[n]:= r;
      found:= true;
    else
      P:= [op(P), seq(ithprime(i),i=np+1..np+20)];
      np:= np+20;
    fi
  od;
od:
seq(A[i],i=1..100); # Robert Israel, Apr 30 2018

cp's OEIS Frontend

A084865 Primes of the form 2x^2 + 3y^2.

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A084864 Numbers that can be written in the form 2*u^2 + 3*v^2, u*v>0.

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A084863 Number of solutions to n = 2*u^2 + 3*v^2, u*v>0.

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A259142 Least prime p of the form n*q^2+(n+1)*r^2 with q and r prime.

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Maple

A084864 Numbers that can be written in the form 2u^2 + 3v^2, u*v>0.

A084863 Number of solutions to n = 2u^2 + 3v^2, u*v>0.

A259142 Least prime p of the form nq^2+(n+1)r^2 with q and r prime.