A381226 a(n) is the number of distinct positive integers that can be obtained by starting with n!, and optionally applying the operations square root, floor, and ceiling, in any order.

1, 2, 4, 6, 7, 8, 8, 9, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset

1,2

Comments

This sequence, A381227, and A381228 arose in connection with the problem of showing that every positive integer can be represented using a single 4. Hans Havermann has pointed out that A139004 is related to this question and has many references. - N. J. A. Sloane, Feb 25 2025

Links

Table of n, a(n) for n=1..68.

Example

For n = 8, 8! = 40320; sqrt(40320) = 200.798..., floor and ceiling give 200 and 201. Sqrt(200) = 14.142..., and floor and ceiling give 14 and 15. From 14 we get 3 and 4; from 3 we get 1 and 2. 15 and 4 give nothing more. In all, we get a(8) = 9 different numbers: 40320, 200, 201, 14, 15, 3, 4, 1, 2.
Note that at each step, we must consider three "parents": if x was a term at the previous step, we get floor(sqrt(x)), sqrt(x), and ceiling(sqrt(x)) as potential parents at the next step.

Prog

(PARI) f(n) = my(t); if(n<4, [1..n], t=sqrtint(n); if(issquare(n), concat(f(t), n), Set(concat([f(t), f(t+1), [n]]))));
a(n) = #f(n!); \\ Jinyuan Wang, Feb 25 2025

Crossrefs

Cf. A139004, A381227-A381229.

Motivated by trying to understand A000319.

Sequence in context: A023835 A272633 A240817 * A174416 A228728 A138888

Adjacent sequences: A381223 A381224 A381225 * A381227 A381228 A381229

Keyword

nonn

Author

N. J. A. Sloane, Feb 24 2025

Extensions

More terms from Jinyuan Wang, Feb 25 2025

Status

approved