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%I #13 May 26 2017 18:58:08
%S 1,1,5,6,9,0,5,1,5,3,7,5,4,0,2,8,9,5,4,5,0,1,3,4,5,8,1,5,5,7,2,3,2,1,
%T 4,6,5,3,5,2,5,5,4,0,2,8,9,4,8,7,9,5,3,6,4,7,0,0,3,9,9,3,8,9,5,9
%N Constant whose Engel Expansion is A007775.
%C This one constant is enough information to uniquely reconstruct A007775.
%C There appears to be a general expression for higher sets of k-rough numbers.
%H B. W. J. Irwin, <a href="https://www.authorea.com/users/5445/articles/144462/_show_article">Constants Whose Engel Expansions are the k-rough Numbers</a>
%F Define an indexing function over the primes and 7^2.
%F P(n) = prime(n) for n<16, 49 for n=16, prime(n-1) for n>16.
%F a = Pi^4*Sum_{k>=0}Sum_{n=1..8} 2^(4-n-8*k)*15^(-n-8*k)/(Prod_{m=1..8} Gamma( P(2+m+n)/30 + k)). - _Benedict W. J. Irwin_, Dec 16 2016
%e 1.15690515375402895450134581557232146535255402894879536470039938959...
%t Prime7[n_] := If[n < 16, Prime[n], If[n == 16, 7^2, Prime[n - 1]]];
%t RealDigits[N[Pi^4*Sum[Sum[2^(4-n-8*k)*15^(-n-8*k)/Product[Gamma[ Prime7[2+m+n]/30+k], {m,1,8}],{n,1,8}],{k,0,Infinity}], 100]][[1]]
%Y Cf. A007775.
%K nonn,cons
%O 1,3
%A _Benedict W. J. Irwin_, Dec 16 2016
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