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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.

%I A381226 #33 Mar 16 2025 12:55:32
%S A381226 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,
%T A381226 14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,16,16,16,16,16,16,16,17,
%U A381226 17,17,17,17,17,17,17,17,17,18,18,18,18,18,18,18,18,18,18,18
%N 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.
%C A381226 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
%e A381226 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.
%e A381226 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.
%o A381226 (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]]))));
%o A381226 a(n) = #f(n!); \\ _Jinyuan Wang_, Feb 25 2025
%Y A381226 Cf. A139004, A381227-A381229.
%Y A381226 Motivated by trying to understand A000319.
%K A381226 nonn
%O A381226 1,2
%A A381226 _N. J. A. Sloane_, Feb 24 2025
%E A381226 More terms from _Jinyuan Wang_, Feb 25 2025