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A338088 Smallest prime numbers which can be represented as x^2 + h*y^2 with x > 0 for every h in the first n idoneal numbers. 2
2, 17, 73, 73, 241, 241, 1009, 1009, 1009, 1009, 1009, 2521, 2521, 2521, 2521, 2521, 8089, 8089, 8089, 8089, 8089, 8089, 19009, 19009, 19009, 19009, 19009, 19009, 53881, 53881, 53881, 53881, 53881, 53881, 53881, 605641, 605641, 605641, 605641, 605641, 605641 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The sequence lists prime numbers, in nondecreasing order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes an increasing number of values of the sequence A000926 (idoneal numbers). See examples.

LINKS

Marco Frigerio, Table of n, a(n) for n = 1..65

EXAMPLE

a(1) = 2 because, for A000926(1) = 1, 2 = 1^2+A000926(1)*1^2.

a(2) = 17 because, considered the first two idoneal numbers, A000926(1) = 1 and A000926(2) = 2, 17 = 1^2+A000926(1)*4^2 = 3^2+A000926(2)*2^2.

The prime 1009 is present in the sequence 5 times because:

a(7) = 15^2+1*28^2 = 19^2+2*18^2 = 31^2+3*4^2 = 15^2+4*14^2 = 17^2+5*12^2 = 25^2+6*12^2 = 1^2+7*12^2, with idoneal numbers up to A000926(7), and also:

a(8) = 19^2+8*9^2,

a(9) = 28^2+9*5^2,

a(10) = 3^2+10*10^2,

a(11) = 31^2+12*2^2,

with idoneal numbers from  A000926(8) to A000926(11).

1083289 is the last term of the sequence since for every idoneal number h there are x, y such that 1083289 = x^2 + h*y^2 and this is the least prime for which this is possible.

PROG

(PARI)

isok(p, u)={for(i=1, #u, my(s=qfbsolve(Qfb(1, 0, u[i]), p)); if(s==0 || s[1]==0, return(0))); 1}

idoneal()={select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848])}

seq()={my(u=idoneal(), v=[1], L=List()); forprime(p=2, oo, if(isok(p, v), listput(L, p); my(k=#v); while(k<#u && isok(p, [u[k+1]]), listput(L, p); k++); if(k==#u, return(Vec(L))); v=u[1..k+1]))}  \\ Andrew Howroyd, Nov 05 2020

CROSSREFS

Cf. A000926.

Sequence in context: A056074 A320685 A268784 * A155715 A307851 A054568

Adjacent sequences:  A338085 A338086 A338087 * A338089 A338090 A338091

KEYWORD

nonn,fini,full

AUTHOR

Marco Frigerio, Oct 09 2020

STATUS

approved

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Last modified October 22 21:20 EDT 2021. Contains 348180 sequences. (Running on oeis4.)