login
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).
3
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
1,1
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
STATUS
approved