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%I #16 Aug 08 2022 14:22:20
%S 1,3,5,10,19,35,64,139,256,536,1061,2095,4169,8282,16517,32903,65646,
%T 131205,262579,525083,1048893,2098826,4195521,8390583,16782032,
%U 33569609,67118347,134229613,268453180,536890474,1073764782,2147523518
%N 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 a(n) is near the (n-1)th power of 2, the difference is A085355.
%F a(n) = Min{x; Floor[Log[2, (x+1)! ]]-Floor[Log[2, x! ]]=n}= Min{x; A084320(x)=n}.
%e 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 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}]
%Y Cf. A067850, A058033, A000142, A000079, A084320, A084420.
%K nonn
%O 1,2
%A _Labos Elemer_, Jun 19 2003
%E Edited and extended by _Robert G. Wilson v_, Jun 24 2003
%E Definition clarified by _Jianing Song_, Aug 08 2022
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