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A269225 Smallest k such that k! > 2^n.

%I A269225 #21 May 22 2025 10:21:44
%S A269225 2,3,3,4,4,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,11,12,12,12,
%T A269225 13,13,13,13,14,14,14,14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,18,
%U A269225 18,19,19,19,19,20,20,20,20,20,21,21,21,21,22,22,22,22,23,23,23,23,23,24,24,24,24,24,25,25,25,25,26,26,26,26,26,27,27,27,27
%N A269225 Smallest k such that k! > 2^n.
%H A269225 Charles R Greathouse IV, <a href="/A269225/b269225.txt">Table of n, a(n) for n = 0..10000</a>
%e A269225 a(7) = 6 because 6! = 720 > 2^7 = 128, but 5! = 120 < 128.
%t A269225 a[n_] := Block[{v=2^n, k=1}, While[++k! <= v]; k]; Array[a, 93, 0] (* _Giovanni Resta_, Jul 11 2016 *)
%t A269225 Module[{nn=30,f},f=Table[{k,k!},{k,nn}];Table[SelectFirst[f,#[[2]]>2^n&],{n,0,100}]][[;;,1]] (* _Harvey P. Dale_, Feb 19 2024 *)
%o A269225 (Python)
%o A269225 def a269225():
%o A269225    k = 1
%o A269225    f = 1
%o A269225    p = 1
%o A269225    n = 0
%o A269225    while True:
%o A269225       while f<=p:
%o A269225          k += 1
%o A269225          f *= k
%o A269225       yield k
%o A269225       p *= 2
%o A269225       n += 1
%o A269225 (PARI) a(n)=localprec(19); my(t=log(2)*n, x=ceil(solve(k=1, n/2+5, lngamma(k+1)-t))); while(x!<=2^n, x++); x \\ _Charles R Greathouse IV_, Jul 12 2016
%Y A269225 Cf. A003070, A067850, A258782.
%K A269225 nonn,easy
%O A269225 0,1
%A A269225 _Christian Perfect_, Jul 11 2016