A084321 Least number k such that between k! and (k+1)! there are n powers of 2 (each interval includes (k+1)! but not k!).

1, 3, 5, 10, 19, 35, 64, 139, 256, 536, 1061, 2095, 4169, 8282, 16517, 32903, 65646, 131205, 262579, 525083, 1048893, 2098826, 4195521, 8390583, 16782032, 33560609, 67118347, 134229613, 268453180, 536890474, 1073764782, 2147523518

Offset: 1

Author

Labos Elemer, Jun 19 2003

Keywords

• nonn

Comments

a(n) is near the (n-1)th power of 2, the difference is A085355.

Examples

			a(3)=5 since between 5!=120 and 6!=720 is the first time 3 powers of 2 arise, namely, 128, 256 and 512.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..750
• Kevin Ryde, C Code

Crossrefs

Cf. A067850, A058033, A000142, A000079, A084320, A084420, A085355.

Programs

C

/* See links */

Mathematica

LogBase2Stirling[n_] := N[ Log[2, 2 Pi n]/2 + n*Log[2, n/E] + Log[2, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)], 64]; k = 1; Do[ While[ Floor[ LogBase2Stirling[k + 1]] - Floor[ LogBase2Stirling[k]] < n, k++ ]; Print[k], {n, 1, 33}]

Formula

a(n) = minimum x for which floor(log_2((x+1)!)) - floor(log_2(x!)) = n.

a(n) = minimum x for which A084320(x) = n.

Extensions

Edited and extended by Robert G. Wilson v, Jun 24 2003

Definition clarified by Jianing Song, Aug 08 2022

a(26) corrected by Kevin Ryde, Apr 25 2024