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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

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)).

a(n) is the least prime that either decomposes or ramifies in the imaginary quadratic field with discriminant D, D = -A003657(n).

The quadratic field with discriminant D = -A003657(n) has class number 1 if and only if a(n) >= (1/4)*A003657(n). If the quadratic field with discriminant D = -A003657(n) has class number 3 then a(n)^2 < (1/4)*A003657(n) < a(n)^3.

For most n, a(n) is relatively small. There are only 86 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)). In fact, if we ignore the first term, the only terms among the first 3043 ones that seem unusually large are a(15) = 11 (with A003657(15) = 43), a(22) = 17 (with A003657(22) = 67), a(52) = 41 (with A003657(52) = 163), a(1147) = 23 (with A003657(1147) = 3763), a(2677) = 23 (with A003657(2677) = 8803) and a(2758) = 23 (with A003657(2758) = 9067).

It seems that this sequence contains 810 terms, the largest being 1154008. In general, it seems that for any t > 0, b(-d) = o(d^t) as -d -> -oo.

For fundamental discriminants -d, we want to determine the size of b(-d), i.e., the size of the smallest prime that decomposes in Q[sqrt(-d)].

Let K = Q[sqrt(-d)], O_K be the ring of integers over K, so O_K is a Dedekind domain. Let E(-d) be the exponent of the ideal class group of O_K (the exponent of a group G is the smallest e > 0 such that x^e = I for all x in G, where I is the group identity).

If Kronecker(-d,p) = 1, it is well-known that p*O_K is the product of two distinct prime ideals of O_K, say, p*O_K = I*I'. By the property of the ideal class group of Q[sqrt(-d)], I^(k*e) must be principal, e = E(-d). Let t*O_K = I^(k*e), then t/p is not an algebraic integer, and the norm of t is p^e. Define f(x,y) = x^2 + (d/4)*y^2 if -d == 0 (mod 4), x^2 + x*y + ((d+1)/4)*y^2 otherwise, it is easy to see f(x,y) = p^(k*e) has integral solutions (x,y) such that gcd(x,y) = 1.

If f(x,y) = p^(k*e) < d, then |y| = 1, so 4*p^(k*e) - d must be a (positive) square. Setting k = 1 gives b(-d) > (d/4)^(1/e) (and furthermore we have: if Kronecker(-d,p) = 1 and p^(k*e) < d, then k = 1, or (p,k,e,d) = (2,2,1,7), (3,2,1,11)).

If E(-d) = 3, then d is in this sequence.

We also have the following observations (not proved):

(a) if e = 2 (i.e., d is in A003644\A014602 = A316743), then b(-d) < d/4;

(b) if e > 2, then b(-d) < sqrt(d/4) (it can be proved by using deeper algebraic number theory that b(-d) < 2*sqrt(d)/Pi).

If these observations are true, this sequence is also the list of d such that b(-d) > (d/4)^(1/3) and d is not in A003644.

Note that 5460 is conjectured to be the largest term in A003644. Therefore, it seems that b(-d) < sqrt(d/4) for all d > 5460; it seems that b(-d) < (d/4)^(1/3) for all d > 1154008.

Among the known terms:

(1) the term d with the largest E(-d) is d = 998328 with E(-d) = 66.

(2) the term d with the largest b(-d) is d = 656755 with b(-d) = 79.

(3) the largest prime is d = 90787 with E(-d) = 23.

If a(n) < (Pi*prime(n)/2)^2 (this occurs for n <= 14), then the ideal class group of Q[sqrt(-d)] necessarily has exponent <= 2. (The exponent of a group G is the smallest e > 0 such that x^e = I for all x in G, where I is the group identity.) See A330221.

It seems that lim_{n->oo} n^t/a(n) = 0 for all t > 0.

The exponent of the ideal class group of Q[sqrt(-a(n))]: 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 26, 16, 36, 22, 38, 24, 16, 36, 104, 388, 104, 288, ...