A109054 Squares and numbers k such that the continued fraction expansion of sqrt(k) is multiplicative.

0, 1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 22, 23, 24, 25, 32, 33, 34, 35, 36, 44, 47, 48, 49, 58, 59, 60, 62, 63, 64, 74, 75, 78, 79, 80, 81, 95, 96, 98, 99, 100, 114, 119, 120, 121, 135, 136, 138, 140, 141, 142, 143, 144, 160, 162, 164, 167, 168, 169, 185, 187, 189, 192

Offset

1,3

Comments

If we consider each square k as having a continued fraction expansion c of all zeros after c(0) = sqrt(k)-1, then the continued fraction expansion of sqrt(k) for each square is trivially multiplicative.

For nonsquares, c(1) must be 1 and so k must satisfy m + 1/2 < sqrt(k) <= m+1, for some integer m.

Links

Table of n, a(n) for n=1..63.

Example

The continued fraction of sqrt(22) is c = (4; 1, 2, 4, 2, 1, 8, ...) = A010126, which is multiplicative with c(2^e) = 2, c(3^e) = 4, c(p^e) = 1 otherwise.

Crossrefs

Union of A000290 and A108575.

Continued fraction expansions: A040001, A010121, A040005, etc.

Sequence in context: A319736 A100452 A004201 * A350690 A363751 A129142

Adjacent sequences: A109051 A109052 A109053 * A109055 A109056 A109057

Keyword

nonn,easy

Author

Mitch Harris, Jun 18 2005

Status

approved