OFFSET
1,2
COMMENTS
a(n) is A145571(n) (a decimal number with digits only from {0,1}) read as base 2 number converted back into decimal notation.
The sequence of digit lengths is [1,1,2,8,37,217,1518,...]
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
FORMULA
a(n) = A145571(n) interpreted as number in binary notation, then converted to decimal notation.
From Wolfdieter Lang, Apr 10 2024: (Start)
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.
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.
(End)
EXAMPLE
a(3)=49, because A145571(3)=110001, and the binary number 110001 translates to 2^5+2^4+2^0=32+16+1 = 49.
MATHEMATICA
a[n_] := FromDigits[RealDigits[Sum[1/10^k!, {k, n}], 10, n!][[1]], 2]; Array[a, 6] (* Robert G. Wilson v, Aug 08 2018 *)
Block[{k = 0}, NestList[#*2^(++k*k!) + 1 &, 1, 5]] (* Paolo Xausa, Jun 27 2024 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Wolfdieter Lang Mar 06 2009
STATUS
approved