OFFSET
1,1
COMMENTS
a(n) is the least prime that decomposes in the imaginary quadratic field with discriminant D, D = -A003657(n).
For most n, a(n) is relatively small. There are only 472 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)).
Also a(n) is the smallest prime p such that the imaginary quadratic field with discriminant D = -A003657(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021
EXAMPLE
Let K = Q[sqrt(-177)] with D = -708 = -A003657(218), we have: 2 and 3 divides 708, (-708/5) = (-708/7) = ... = (-708/29) = -1 and (-708/31) = +1, so 2 and 3 ramify in K, 5, 7, ..., 29 remain inert in K and 31 decomposes in K, so a(218) = 31.
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
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 22 2019
STATUS
approved