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%I #21 Nov 07 2022 02:10:07
%S 1,5,10,24,41,52,152,162,364,726,1150,2451,4626,9847,18131,36016,
%T 71709,143848,276769,551730,1086371,2158296,4297353,8607525,17159741,
%U 34152001,68194361,136211839,271350906,541199486,1084811069,2165421369,4331203801,8643518017,17303787585
%N Positions of the first n-bit number to appear in Van Eck's sequence (A181391).
%C Binary version of the concept behind A358180.
%e First terms written in binary, substituting "." for 0 to enhance the pattern of 1's.
%e n a(n) a(n)_2
%e -------------------------------------
%e 1 1 1
%e 2 5 1.1
%e 3 10 1.1.
%e 4 24 11...
%e 5 41 1.1..1
%e 6 52 11.1..
%e 7 152 1..11...
%e 8 162 1.1...1.
%e 9 364 1.11.11..
%e 10 726 1.11.1.11.
%e 11 1150 1...111111.
%e 12 2451 1..11..1..11
%e 13 4626 1..1....1..1.
%e 14 9847 1..11..111.111
%e 15 18131 1...11.11.1..11
%e 16 36016 1...11..1.11....
%e 17 71709 1...11......111.1
%e 18 143848 1...11...1111.1...
%e 19 276769 1....111..1..1....1
%e 20 551730 1....11.1.11..11..1.
%e 21 1086371 1....1..1..111.1...11
%e 22 2158296 1.....111.111.11.11...
%e 23 4297353 1.....11..1..1.1...1..1
%e 24 8607525 1.....11.1.1.111..1..1.1
%e etc.
%t nn = 2^20; q[_] = False; q[0] = True; a[_] = 0; c[_] = -1; c[0] = 2; m = 1; {1}~Join~Rest@ Reap[Do[j = c[m]; k = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#], Sow[n]; q[#] = True] & @ IntegerLength[k, 2], {n, 3, nn}] ][[-1, -1]]
%o (Python)
%o from itertools import count
%o def A358259(n):
%o b, bdict, k = 0, {0:(1,)},1<<n-1 if n > 1 else 0
%o for m in count(2):
%o if b >= k:
%o return m-1
%o if len(l := bdict[b]) > 1:
%o b = m-1-l[-2]
%o if b in bdict:
%o bdict[b] = (bdict[b][-1],m)
%o else:
%o bdict[b] = (m,)
%o else:
%o b = 0
%o bdict[0] = (bdict[0][-1],m) # _Chai Wah Wu_, Nov 06 2022
%Y Cf. A181391, A358168, A358180, A358258.
%K nonn,base
%O 1,2
%A _Michael De Vlieger_, Nov 05 2022
%E a(30)-a(34) from _Chai Wah Wu_, Nov 06 2022
%E a(35) from _Martin Ehrenstein_, Nov 07 2022
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