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

Author

N. J. A. Sloane, Feb 24 2025

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A139004, A381227-A381229.

Motivated by trying to understand A000319.

Programs

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

Extensions

More terms from Jinyuan Wang, Feb 25 2025