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A145572 Numerators of partial sums for Liouville's constant, read as base 2 (binary) numbers.

%I A145572 #29 Feb 02 2025 09:30:07
%S A145572 1,3,49,12845057,1017690263500988729456314874071089153,
%T A145572 4222921592695952872362526736376161058920018764920519780147745963811744865992371113095993596088044297100172572224585271942341064532181870606866447799704872724575357044373908131956500952542608981420222196042850818326529
%N A145572 Numerators of partial sums for Liouville's constant, read as base 2 (binary) numbers.
%C A145572 a(n) is A145571(n) (a decimal number with digits only from {0,1}) read as base 2 number converted back into decimal notation.
%C A145572 The sequence of digit lengths is 1,1,2,8,37,217,1518,... (see A317873).
%C A145572 This sequence gives the numerators of the partial sums for the constant A092874 (called there "binary" Liouville number). See the B(n) formula below. - _Wolfdieter Lang_, Apr 10 2024
%F A145572 a(n) = A145571(n) interpreted as number in binary notation, then converted to decimal notation.
%F A145572 From _Wolfdieter Lang_, Apr 10 2024: (Start)
%F A145572 a(n) = Sum_{j=0..n} 2^(n! - j!) = 2^(n!)*B(n) = numerator(B(n)), where B(n) := Sum_{j=1..n} 1/2^(j!), for n >= 1 (Proof from the positions of 1 in A145571).
%F A145572 a(1) = 1, and a(n) = a(n-1)*2^z(n) + 1, where z(n) = n! - (n-1)! = A001563(n-1), for n >= 2.
%F A145572 (End)
%e A145572 a(3)=49, because A145571(3)=110001, and the binary number 110001 translates to 2^5+2^4+2^0=32+16+1 = 49.
%t A145572 a[n_] := FromDigits[RealDigits[Sum[1/10^k!, {k, n}], 10, n!][[1]], 2]; Array[a, 6] (* _Robert G. Wilson v_, Aug 08 2018 *)
%t A145572 Block[{k = 0}, NestList[#*2^(++k*k!) + 1 &, 1, 5]] (* _Paolo Xausa_, Jun 27 2024 *)
%Y A145572 Cf. A001563, A092874, A145571 (numerators of approximations for Liouville's number).
%Y A145572 Cf. A317873.
%K A145572 nonn,base,easy
%O A145572 1,2
%A A145572 _Wolfdieter Lang_ Mar 06 2009