%I #7 Apr 26 2018 13:10:49

%S 1,1,1,4,0,4,6,3,5,1,0,3,8,0,0,3,0,4,9,6,1,4,9,9,4,2,3,6,2,0,0,1,7,7,

%T 2,4,7,5,6,5,1,4,3,1,6,5,5,5,8,3,8,9,0,2,3,0,6,5,1,1,1,4,5,4,0,1,4,8,

%U 1,8,6,8,5,5,4,9,2,1,6,4,9,6,1,0,5,8,0,3,4,5,4,6,7,1,5,3,0,9,8,5,3,4,7,8,7,7,2,7,0,3,7,2,3,0,1,8,4,2,1,7

%N Decimal expansion of constant A = Sum_{n>=1} 1 / (2^n - 1)^n.

%F This constant may be defined by the following expressions.

%F (1) A = Sum_{n>=1} 1 / (2^n - 1)^n.

%F (2) A = Sum_{n>=1} (2^n + 1)^(n-1) / 2^(n^2).

%F (3) A = Sum_{n>=1} A143862(n)/2^n where A143862(n) = Sum_{d|n} binomial(n/d-1, d-1) for n>=1.

%e Constant A = 1.1140463510380030496149942362001772475651431655583890...

%e This constant equals the sum of the following infinite series.

%e (1) A = 1 + 1/3^2 + 1/7^3 + 1/15^4 + 1/31^5 + 1/63^6 + 1/127^7 + 1/255^8 + 1/511^9 + 1/1023^10 + 1/2047^11 + 1/4095^12 + 1/8191^13 + 1/16383^14 + ...

%e Also,

%e (2) A = 1/2 + 5/2^4 + 9^2/2^9 + 17^3/2^16 + 33^4/2^25 + 65^5/2^36 + 129^6/2^49 + 257^7/2^64 + 513^8/2^81 + 1025^9/2^100 + 2049^10/2^121 + 4097^11/2^144 + ...

%e Expressed in terms of powers of 1/2, we have

%e (3) A = 1/2 + 1/2^2 + 1/2^3 + 2/2^4 + 1/2^5 + 3/2^6 + 1/2^7 + 4/2^8 + 2/2^9 + 5/2^10 + 1/2^11 + 9/2^12 + 1/2^13 + 7/2^14 + 7/2^15 + 9/2^16 + 1/2^17 + 19/2^18 + 1/2^19 + 14/2^20 + 16/2^21 + 11/2^22 + ... + A143862(n)/2^n + ...

%e DECIMAL EXPANSION TO 1000 DIGITS:

%e A = 1.11404635103800304961499423620017724756514316555838\

%e 90230651114540148186855492164961058034546715309853\

%e 47877270372301842177485420929460338909110702521744\

%e 10383049196253371844115566456211414378684927895066\

%e 60974873819605352670009454376709247947228660654797\

%e 93238935770616752469239881642090329202510251771440\

%e 45431299113929370687739805515426044704234082381940\

%e 40977172853717815297745712948744536180513363052564\

%e 03854647492812806063479313722184475876462500578835\

%e 52045304926771113275210795841642087115096877536105\

%e 78958100347787242164699010158545554930990338272655\

%e 43040184293715344496344121360156193744995971261933\

%e 73889789233802551892919983961109429243561889220300\

%e 14247648527081291849257683339708248465152426049641\

%e 37604659963590944969427064766226893075229683791387\

%e 71510378240823470806036647280652258639308495696013\

%e 02313861338203801779994141266165775176960964298159\

%e 94404826055321122142831544652111697011832111972164\

%e 78072293762844331943953407090067036379926709332730\

%e 54952123380671191947076262400344800454366565708630...

%Y Cf. A303561 (binary), A302765, A143862.

%K nonn,cons

%O 1,4

%A _Paul D. Hanna_, Apr 26 2018