A346420 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

2, 2, 1, 4, 6, 2, 2, 6, 2, 1, 8, 2, 2, 2, 78, 1, 1, 84, 10, 4, 2, 2, 6, 1, 4, 546, 2, 8, 12, 2, 2, 12, 8, 2, 10, 4, 1062, 6, 1, 7176, 14, 2, 2, 14, 1, 1, 4, 24, 8, 40, 26, 138, 2, 5, 16, 6, 2, 16, 11934, 2, 3, 60, 826, 2, 250, 10, 2, 78, 1, 12, 18, 1, 2, 18, 1, 1, 2244, 6, 84

Offset: 1

Author

Sean A. Irvine

Keywords

• nonn

Comments

The radical part is actually sqrt(A007913(A000037(n))) where A007913(m) is the squarefree part of m. - Michel Marcus, Jun 26 2020

How does this sequence differ from A048942? The definitions of both sequences are identical, but the second comment in A048942 states the terms differ from n = 14 onwards. - Felix Fröhlich, Jun 16 2022

Links

• S. R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]
• Eric Weisstein's World of Mathematics, Fundamental Unit.

Crossrefs

Cf. A000037, A007913, A346419, A048941, A048942.

Programs

PARI

f(n) = {if (issquare(n), return (0)); if (!issquarefree(n), m = core(n), m = n); my(u = abs(2*polcoeff(lift(bnfinit(x^2-m, 1).fu[1]), 0))); if (u^2==1, return (1)); if (u^2==4, return (sqrtint((u^2+4)/m));); if (u^2 < 4, return((u^2+4)/n)); my(v2 = [(u^2-4)/m, (u^2+4)/m]); sqrtint(vecmin(select(x->denominator(x)==1, v2)));}
lista(nn) = apply(f, select(x->!issquare(x), [1..nn])); \\ Michel Marcus, Jun 25 2020; corrected Jun 16 2022