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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!).

%I A084321 #22 Apr 25 2024 10:22:07
%S A084321 1,3,5,10,19,35,64,139,256,536,1061,2095,4169,8282,16517,32903,65646,
%T A084321 131205,262579,525083,1048893,2098826,4195521,8390583,16782032,
%U A084321 33560609,67118347,134229613,268453180,536890474,1073764782,2147523518
%N 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!).
%C A084321 a(n) is near the (n-1)th power of 2, the difference is A085355.
%H A084321 Kevin Ryde, <a href="/A084321/b084321.txt">Table of n, a(n) for n = 1..750</a>
%H A084321 Kevin Ryde, <a href="/A084321/a084321.c.txt">C Code</a>
%F A084321 a(n) = minimum x for which floor(log_2((x+1)!)) - floor(log_2(x!)) = n.
%F A084321 a(n) = minimum x for which A084320(x) = n.
%e A084321 a(3)=5 since between 5!=120 and 6!=720 is the first time 3 powers of 2 arise, namely, 128, 256 and 512.
%t A084321 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}]
%o A084321 (C) /* See links */
%Y A084321 Cf. A067850, A058033, A000142, A000079, A084320, A084420, A085355.
%K A084321 nonn
%O A084321 1,2
%A A084321 _Labos Elemer_, Jun 19 2003
%E A084321 Edited and extended by _Robert G. Wilson v_, Jun 24 2003
%E A084321 Definition clarified by _Jianing Song_, Aug 08 2022
%E A084321 a(26) corrected by _Kevin Ryde_, Apr 25 2024