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A352920 Values of A109812(k) where k/A109812(k) reaches a new high point.

%I A352920 #19 Apr 26 2022 13:00:04
%S A352920 1,3,6,7,31,63,127,511,4093,4094,4095,16383,32767,262143
%N A352920 Values of A109812(k) where k/A109812(k) reaches a new high point.
%C A352920 The corresponding values of k are given in A352919.
%C A352920 This is a subset of A352336.
%C A352920 It is not necessary for a term of this sequence to be of the form 2^k - 1: there may be a zero close to the end of the binary expansion.
%C A352920 It appears that n/A109812(n) is unbounded. The reasoning behind this is as follows.
%C A352920 Consider terms A109812(k) that are the form 2^i - 1 (see the Examples section).
%C A352920 For such k, we necessarily have
%C A352920 a(k+1) = p(i)*2^i and a(k-1) = m(i)*2^i,
%C A352920 with integers p(i) and m(i). Let r(i) = max(p(i), m(i)).
%C A352920 Taking A109812(0) = 0, we have the following values:
%C A352920   i : 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18
%C A352920 p(i): 1 2 3 4 5 7 7 9  9 11 12 13 13 15 15 17 17 19 [A352921]
%C A352920 m(i): 0 1 4 3 6 6 8 8 10 10 11 14 14 16 18 18 18 20 [A352922]
%C A352920 r(i): 1 2 4 4 6 7 8 9 10 11 12 14 14 16 18 18 18 20 [A352923]
%C A352920 for i < 19.  Furthermore, from the graphs in the entry A109812 it appears that r(19) = 21, r(20) = 22, r(21) = 22, r(22) = 24. The corresponding four values of k/aA109812(k) are, approximately, 6.42199, 6.80074, 6.88852, 7.39979.
%C A352920 This suggests the following conjecture:
%C A352920 Conjecture: r(k) > k for all k > 4.
%C A352920 Combining this with the conjecture that A109812(k)/k is bounded (see A352919 and A352920), we have:
%C A352920 Conjecture: k/A109812(k) is unbounded.
%e A352920 Let c(k) denote A109812(k). The first 14 record high-points of k/c(k) are as follows:
%e A352920 [k/c(k), k, c(k), "binary(c(n))"]
%e A352920 [1.000000000 1 1 "1"]
%e A352920 [1.333333333 4 3 "11"]
%e A352920 [1.500000000 9 6 "110"]
%e A352920 [2.285714286 16 7 "111"]
%e A352920 [2.451612903 76 31 "11111"]
%e A352920 [2.571428571 162 63 "111111"]
%e A352920 [3.291338583 418 127 "1111111"]
%e A352920 [3.702544031 1892 511 "111111111"]
%e A352920 [4.665037870 19094 4093 "111111111101"]
%e A352920 [4.713727406 19298 4094 "111111111110"]
%e A352920 [4.898412698 20059 4095 "111111111111"]
%e A352920 [5.167124458 84653 16383 "11111111111111"]
%e A352920 [5.327494125 174566 32767 "111111111111111"]
%e A352920 [6.439611205 1688099 262143 "111111111111111111"]
%e A352920 The values of k and c(k) form A352919 and the present sequence.
%Y A352920 Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.
%K A352920 nonn,more
%O A352920 1,2
%A A352920 _David Broadhurst_, Aug 17 2022 (entry created by _N. J. A. Sloane_, Apr 23 2022)