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Decimal expansion of 1 / 1^1 + 1 / (1^1 * 2^2) + 1 / (1^1 * 2^2 * 3^3) + 1 / (1^1 * 2^2 * 3^3 * 4^4) + ...
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%I #20 Oct 07 2023 17:14:41

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

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

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

%N Decimal expansion of 1 / 1^1 + 1 / (1^1 * 2^2) + 1 / (1^1 * 2^2 * 3^3) + 1 / (1^1 * 2^2 * 3^3 * 4^4) + ...

%C This constant is irrational (Stucky, 2016). - _Amiram Eldar_, Apr 30 2023

%H Joshua Stucky, <a href="https://www.kappamuepsilon.org/Pentagon/Vol_75_Num_2_Spring_2016.pdf">Several Theorems Concerning the Irrationality of Certain Infinite Series Involving Tetrations</a>, The Pentagon, Vol. 75, No. 2 (2016), pp. 4-9.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>.

%e 1.2592954398150628876913596498837350926311457518421...

%t With[{m = 105}, RealDigits[N[Sum[1/Hyperfactorial[n], {n, 1, Infinity}], m + 2], 10, m][[1]]] (* _Amiram Eldar_, Apr 30 2023 *)

%t RealDigits[Total[Table[1/Times@@(Range[n]^Range[n]),{n,30}]],10,120][[1]] (* _Harvey P. Dale_, Oct 07 2023 *)

%Y Cf. A000312, A002109, A073009, A083648, A091131, A137986, A287013, A347352.

%K nonn,cons

%O 1,2

%A _Ilya Gutkovskiy_, Aug 28 2021