A066173 Self-reciprocating sequence: the integer part of powers of the reciprocal sum.

1, 3, 5, 9, 17, 31, 55, 99, 176, 313, 557, 990, 1759, 3125, 5553, 9866, 17531, 31149, 55346, 98339, 174729, 310457, 551617, 980109, 1741450, 3094195, 5497739, 9768336, 17356295, 30838517, 54793613, 97356822, 172982767, 307354297

Offset

1,2

Comments

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.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

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.

The constant S = Sum_{n>=1} 1/a(n) is more precisely given by:

S = 1.7767914254 8765842099 7295125934 3751657100 4017014991 1002131974 4535225732 9321570657 9706460392 2109445017 4890160620 5702665489 ... (cf. A195202).

Example

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 +...

Mathematica

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 *)

Crossrefs

Cf. A195202 (constant).

Sequence in context: A143373 A282184 A102475 * A114322 A000213 A074858

Adjacent sequences: A066170 A066171 A066172 * A066174 A066175 A066176

Keyword

easy,nice,nonn

Author

Paul D. Hanna, Dec 14 2001

Status

approved