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A309736 a(1) = 1, and for any n > 1, a(n) is the least k > 0 such that the binary representation of n^k starts with "10".

%I A309736 #14 Feb 16 2025 08:33:55
%S A309736 1,1,2,1,1,2,3,1,1,1,1,2,2,3,5,1,1,1,1,1,1,1,1,2,2,2,2,3,3,5,10,1,1,1,
%T A309736 1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,5,6,10,19,1,1,1,1,
%U A309736 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A309736 a(1) = 1, and for any n > 1, a(n) is the least k > 0 such that the binary representation of n^k starts with "10".
%C A309736 The sequence is well defined; for any n > 0:
%C A309736 - if n is a power of 2, then a(n) = 1,
%C A309736 - if n is not a power of 2, then log_2(n) is irrational,
%C A309736   hence the function k -> frac(k * log_2(n)) is dense in the interval [0, 1]
%C A309736   according to Weyl's criterion,
%C A309736   so for some k > 0, k*log_2(n) = m + 1 + e where m is a positive integer
%C A309736                                               and 0 <= e < log_2(3) - 1 < 1,
%C A309736 - hence 2 * 2^m <= n^k < 3 * 2^m and a(n) <= k, QED.
%H A309736 Rémy Sigrist, <a href="/A309736/a309736.png">Scatterplot of (x, y) such that the binary representation of x^y starts with "10" and x = 2..1024 and y = 1..1024</a>
%H A309736 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WeylsCriterion.html">Weyl's Criterion</a>
%F A309736 a(n) = 1 iff n belongs to A004754.
%F A309736 a(2*n) = a(n).
%F A309736 A090996(n^a(n)) = 1.
%e A309736 For n = 7:
%e A309736 - the first powers of 7, in decimal as well as in binary, are:
%e A309736     k  7^k  bin(7^k)
%e A309736     -  ---  ---------
%e A309736     1    7        111
%e A309736     2   49     110001
%e A309736     3  343  101010111
%e A309736 - hence a(7) = 3.
%o A309736 (PARI) a(n) = { my (nk=n); for (k=1, oo, if (binary(2*nk)[2]==0, return (k), nk *= n)) }
%Y A309736 Cf. A004754, A090996, A098174.
%K A309736 nonn,base
%O A309736 1,3
%A A309736 _Rémy Sigrist_, Aug 14 2019