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Number of B-periodic numbers of bit pseudo-length n.
0

%I #10 Apr 14 2022 15:02:53

%S 1,0,4,0,7,3,16,0,37,0,64,18,127,0,283,0,517,66,1024,0,2167,15,4096,

%T 255,8197,0,16906,0,32767,1026,65536,78,133087,0,262144,4098,524407,0,

%U 1056730,0,2097157,16635,4194304,0,8421247,63,16777711,65538,33554437,0

%N Number of B-periodic numbers of bit pseudo-length n.

%C For the definition of "B-periodic numbers" and "bit pseudo-length", see Dobeš, Kureš, 2010, p. 294. The first few terms are given in the table on p. 295.

%C The sequence counts periodic binary numbers of length n where the least-significant bit is 0 (see Dobeš, Kureš, 2010, p. 294).

%H J. Dobeš and M. Kureš, <a href="http://hdl.handle.net/10525/1595">Search for Wieferich Primes through the use of Periodic Binary Strings</a>, Serdica Journal of Computing, Vol. 4, No. 3 (2010), 293-300.

%F a(n) = (2^n - c(n) - 2)/2, where c(n) = Sum_{d|n} A008683(d)*2^(n/d).

%e For n = 6: The B-periodic numbers of bit pseudo-length 6 are 101010, 100100, 010010 and 110110, so a(6) = 4.

%o (PARI) c(n) = sumdiv(n, d, moebius(d)*2^(n/d))

%o a(n) = (2^n - c(n) - 2)/2

%Y Cf. A008683, A027375, A242138, A242139.

%K nonn,easy,base

%O 4,3

%A _Felix Fröhlich_, Apr 07 2022