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%I #5 Jul 05 2021 02:36:39
%S 1,5,6,2,7,6,0,9,3,4,4,4,6,9,3,7,4,5,5,0,1,4,2,2,7,3,0,8,4,8,9,5,3,0,
%T 3,0,1,9,1,3,8,7,6,8,8,9,0,8,2,0,4,0,0,1,5,0,4,8,3,3,3,6,8,0,8,5,8,4,
%U 1,0,8,5,3,7,9,2,5,4,1,2,0,5,2,7,9,4,5,4,4,6,2,0,0,6,2,3,0,4,7,7
%N Decimal expansion of Sum_{k>=1} 1/A053657(k).
%C This constant is irrational (Mingarelli, 2013).
%H Angelo B. Mingarelli, <a href="http://nntdm.net/volume-19-2013/number-4/43-76/">Abstract factorials</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, No. 4 (2013), pp. 43-76 (see p. 60); <a href="https://arxiv.org/abs/0705.4299">arXiv preprint</a>, arXiv:0705.4299 [math.NT], 2007-2012.
%e 1.56276093444693745501422730848953030191387688908204...
%t f[n_] := Product[p^Sum[Floor[(n - 1)/((p - 1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; RealDigits[Sum[1/f[n], {n, 1, 100}], 10, 100] [[1]]
%Y Cf. A053657.
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, Jul 02 2021