A306541 - OEIS

A306541

The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

2, 5, 2, 2, 3, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 7, 2, 2, 2, 3, 2, 3, 2, 2, 7, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 3, 2, 2, 13, 2, 3, 2, 2, 2, 2, 7, 2, 2, 3, 2, 3, 2, 2, 5, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2

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OFFSET

1,1

COMMENTS

a(n) is the least prime that either decomposes or ramifies in the real quadratic field with discriminant D, D = A003658(n).

For most n, a(n) is relatively small. There are only 86 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).

LINKS

Table of n, a(n) for n=1..88.

EXAMPLE

Let K = Q[sqrt(293)] with D = 293 = A003658(90), we have: (293/2) = (293/3) = ... = (293/13) = -1 and (293/17) = +1, so 2, 3, 5, 7, 11 and 13 remain inert in K and 17 decomposes in K, so a(90) = 17.

PROG

(PARI) b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))

for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

CROSSREFS

Cf. A003658.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); this sequence, A306542 (the least prime that decomposes or ramifies).

Sequence in context: A152765 A327867 A286664 * A242242 A171937 A065291

Adjacent sequences: A306538 A306539 A306540 * A306542 A306543 A306544

KEYWORD

nonn

AUTHOR

Jianing Song, Feb 22 2019

STATUS

approved