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A306537
The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).
3
2, 11, 7, 11, 3, 2, 5, 5, 3, 5, 2, 3, 3, 2, 5, 7, 5, 2, 7, 3, 2, 5, 2, 3, 13, 3, 3, 2, 7, 7, 2, 5, 5, 2, 3, 2, 7, 3, 2, 3, 3, 2, 13, 5, 2, 5, 11, 5, 3, 2, 7, 11, 3, 13, 2, 3, 3, 2, 11, 2, 7, 2, 5, 3, 2, 11, 2, 3, 5, 3, 3, 2, 5, 13, 2, 13, 2, 3, 2, 5, 2, 3, 5, 2
OFFSET
1,1
COMMENTS
a(n) is the least prime that decomposes in the real quadratic field with discriminant D, D = A003658(n).
For most n, a(n) is relatively small. There are only 459 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).
Also a(n) is the smallest prime p such that the real quadratic field with discriminant D = A003658(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021
EXAMPLE
Let K = Q[sqrt(635)] with D = 2540 = A003658(774), we have: 2 and 5 divides 2540, (2540/3) = (2540/7) = ... = (2540/37) = -1 and (2540/41) = +1, so 2 and 5 ramify in K, 3, 7, ..., 37 remain inert in K and 41 decomposes in K, so a(774) = 41.
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); this sequence, A306538 (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).
Sequence in context: A285866 A165768 A323783 * A170873 A224210 A344776
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 22 2019
STATUS
approved