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

1,2

Comments

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

Links

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

Kevin Ryde, C Code

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.

Example

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

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}]

Prog

(C) /* See links */

Crossrefs

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

Sequence in context: A125750 A018168 A320921 * A323812 A270715 A291735

Adjacent sequences: A084318 A084319 A084320 * A084322 A084323 A084324

Keyword

nonn

Author

Labos Elemer, Jun 19 2003

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

Status

approved