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%I #12 Oct 13 2015 18:17:31
%S 1,3,5,9,17,31,55,99,176,313,557,990,1759,3125,5553,9866,17531,31149,
%T 55346,98339,174729,310457,551617,980109,1741450,3094195,5497739,
%U 9768336,17356295,30838517,54793613,97356822,172982767,307354297
%N Self-reciprocating sequence: the integer part of powers of the reciprocal sum.
%C This sequence and its reciprocal sum are unique: there exists only one self-reciprocating sequence whose terms are exactly equal to the integer part of the powers of the sum of the reciprocal terms of the same sequence.
%H Paul D. Hanna, <a href="/A066173/b066173.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) = floor[S^n], where S=1.776791425488... = Sum 1/a(k), k=1, 2, 3, ... The n-th term of the sequence is the integer part of the n-th power of the sum of the infinite series of reciprocal terms of this same sequence.
%F The constant S = Sum_{n>=1} 1/a(n) is more precisely given by:
%F S = 1.7767914254 8765842099 7295125934 3751657100 4017014991 1002131974 4535225732 9321570657 9706460392 2109445017 4890160620 5702665489 ... (cf. A195202).
%e 1=[S], 3=[S^2], 5=[S^3], 9=[S^4], 17=[S^5], 31=[S^6], 55=[S^7], ... where S=1/1 + 1/3 + 1/5 + 1/9 + 1/17 + 1/31 + 1/55 + 1/99 + 1/176 +...
%t digits = 200; Clear[s]; s[m_] := s[m] = x /. FindRoot[x==Sum[1/Floor[x^n], {n, 1, m}], {x, 3/2, 2}, WorkingPrecision -> digits]; s[0] = 0; dm = 100; s[m = dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], m = m + dm; Print[m, " terms"]]; S = s[m]; a[n_] := Floor[S^n]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 13 2015 *)
%Y Cf. A195202 (constant).
%K easy,nice,nonn
%O 1,2
%A _Paul D. Hanna_, Dec 14 2001
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