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A145572
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Numerators of partial sums for Liouville's constant, read as base 2 (binary) numbers.
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3
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1, 3, 49, 12845057, 1017690263500988729456314874071089153, 4222921592695952872362526736376161058920018764920519780147745963811744865992371113095993596088044297100172572224585271942341064532181870606866447799704872724575357044373908131956500952542608981420222196042850818326529
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OFFSET
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1,2
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COMMENTS
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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 constannt A092874 (called there "binary" Liouville number. See the B(n) formula below. Wolfdieter Lang, Apr 10 2024
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LINKS
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FORMULA
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a(n)=A145571(n) interpreted as number in binary notation, then converted to decimal notation.
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)
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EXAMPLE
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a(3)=49, because A145571(3)=110001, and the binary number 110001 translates to 2^5+2^4+2^0=32+16+1 = 49.
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MATHEMATICA
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a[n_] := FromDigits[RealDigits[Sum[1/10^k!, {k, n}], 10, n!][[1]], 2]; Array[a, 6] (* Robert G. Wilson v, Aug 08 2018 *)
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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