

A344073


Smallest number k such that C(k) is the cyclic group of order n, where C(D) is the class group of the quadratic field with discriminant D; or 0 if no such k exists.


3



3, 15, 23, 39, 47, 87, 71, 95, 199, 119, 167, 327, 191, 215, 239, 407, 383, 335, 311, 776, 431, 591, 647, 695, 479, 551, 983, 831, 887, 671, 719, 791, 839, 1079, 1031, 959, 1487, 1199, 1439, 1271, 1151, 1959, 1847, 1391, 1319, 2615, 3023, 1751, 1511, 1799
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OFFSET

1,1


COMMENTS

Conjecture 1: a(n) > 0 for all n;
Conjecture 2: a(n) = o(n^2).
What's the next even term after a(20) = 776 and a(104) = 14024?


LINKS



FORMULA

For odd n, if a(n) > 0, then a(n) >= A060649(n). The smallest odd n such that the inequality is strict is n = 243.
For even n, if a(n) > 0, A060649(n) > 0 and A344072(n/2) > 0, then a(n) >= min{A060649(n), A344072(n/2)/4}. Assuming Conjecture 2 in A344072, we have a(n) >= A060649(n). The smallest n == 2 (mod 4) such that the inequality is strict is n = 342.


EXAMPLE

The smallest k such that c(k) = C_12 is k = 327, so a(12) = 327.
The smallest k such that c(k) = C_16 is k = 407, so a(16) = 407.
The smallest k such that c(k) = C_20 is k = 776, so a(20) = 776.
The smallest k such that c(k) = C_243 is k = 38231, so a(243) = 38231.


PROG

(PARI) a(n) = if(n==1, 3, my(d=3); while(!isfundamental(d)  quadclassunit(d)[2]!=[n], d++); d)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



