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A332520 Fixed points of A331364.

%I A332520 #57 May 28 2025 20:11:15
%S A332520 0,1,2,3,4,5,8,10,12,15,16,17,32,34,48,51,64,68,80,85,128,136,160,170,
%T A332520 192,204,240,255,256,257,512,514,768,771,1024,1028,1280,1285,2048,
%U A332520 2056,2560,2570,3072,3084,3840,3855,4096,4112,4352,4369,8192,8224,8704
%N A332520 Fixed points of A331364.
%C A332520 These are the numbers with at most one kind of nonzero digit in any base of the form 2^2^k (with k >= 0).
%C A332520 If k belongs to the sequence, then A001196(k) also belongs to the sequence, and conversely.
%C A332520 For any positive term m:
%C A332520 - the number of runs of consecutive 1's in the binary representation of m is a power of 2,
%C A332520 - the runs of consecutive 1's in the binary representation of m have all the same length, a power of 2.
%C A332520 Apparently, for any k >= 0, there are A001316(k) nonzero terms with 1+k binary digits.
%H A332520 Rémy Sigrist, <a href="/A332520/b332520.txt">Table of n, a(n) for n = 1..6562</a>
%H A332520 Rémy Sigrist, <a href="/A332520/a332520.png">Binary plot of the terms < 2^2^10</a>
%H A332520 Rémy Sigrist, <a href="/A332520/a332520.gp.txt">PARI program for A332520</a>
%e A332520 The first terms, alongside their binary representation, are:
%e A332520   n   a(n)  bin(a(n))
%e A332520   --  ----  ---------
%e A332520    1     0          0
%e A332520    2     1          1
%e A332520    3     2         10
%e A332520    4     3         11
%e A332520    5     4        100
%e A332520    6     5        101
%e A332520    7     8       1000
%e A332520    8    10       1010
%e A332520    9    12       1100
%e A332520   10    15       1111
%e A332520   11    16      10000
%e A332520   12    17      10001
%e A332520   13    32     100000
%e A332520   14    34     100010
%e A332520   15    48     110000
%e A332520   16    51     110011
%o A332520 (PARI) \\ See Links section.
%Y A332520 Cf. A001196, A001316, A331364.
%K A332520 nonn,base
%O A332520 1,3
%A A332520 _Rémy Sigrist_, Jun 24 2020