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A206578
The least number with exactly n ones in the continued fraction of its square root.
6
2, 3, 14, 7, 13, 91, 43, 115, 94, 819, 133, 1075, 211, 1219, 309, 871, 421, 1147, 244, 3427, 478, 2575, 991, 8791, 604, 3799, 886, 5539, 1381, 8851, 1279, 7303, 1561, 19519, 1759, 10339, 1831, 12871, 2038, 13771, 1999, 8611, 1516, 15871, 2731, 20875, 1726
OFFSET
0,1
COMMENTS
It appears that only the odd-numbered terms 3 and 7 are prime; all other primes occur at even-numbered terms 0, 4, 6, 12, 16, 22, 28, 30, 34, ... In terms 0 to 1000, there are 268 primes and 632 semiprimes.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
MATHEMATICA
nn = 50; zeros = nn; t = Table[0, {nn}]; k = 2; While[zeros > 0, If[! IntegerQ[Sqrt[k]], cnt = Count[ContinuedFraction[Sqrt[k]][[2]], 1]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = k; zeros--]]; k++]; Join[{2}, t]
PROG
(Python)
from sympy import continued_fraction_periodic
def A206578(n):
m = 1
while True:
s = continued_fraction_periodic(0, 1, m)[-1]
if isinstance(s, list) and s.count(1) == n:
return m
m += 1 # Chai Wah Wu, Jun 12 2017
CROSSREFS
Cf. A013647-A013650 (0-3), A020440-A020446 (4-10), A031779-A031868 (11-100).
Cf. A206582 (n twos), A206583 (n threes), A206584 (n fours), A206585 (n fives).
Sequence in context: A287858 A288648 A287912 * A056435 A244295 A032806
KEYWORD
nonn
AUTHOR
T. D. Noe, Feb 24 2012
STATUS
approved