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A003158 A self-generating sequence (see Comments in A003156 for the definition).

%I A003158 M1734 #35 Apr 21 2025 08:37:04
%S A003158 2,7,10,13,18,23,28,31,34,39,42,45,50,53,56,61,66,71,74,77,82,87,92,
%T A003158 95,98,103,108,113,116,119,124,127,130,135,138,141,146,151,156,159
%N A003158 A self-generating sequence (see Comments in A003156 for the definition).
%C A003158 Numbers not of the form Sum_{i>=2} e_i*A001045(i), with e(i) = 0 or 1.
%C A003158 Indices of b in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - _Philippe Deléham_, Apr 12 2004
%D A003158 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003158 Alfred Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 67.
%H A003158 L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz3-a.pdf">Representations for a special sequence</a>, Fibonacci Quarterly 10.5 (1972), 499-518, 550.
%F A003158 a(n) = A003157(n) - 1 = A079523(n) + n. - _Philippe Deléham_, Feb 22 2004
%o A003158 (Python)
%o A003158 def A003158(n):
%o A003158     def bisection(f,kmin=0,kmax=1):
%o A003158         while f(kmax) > kmax: kmax <<= 1
%o A003158         kmin = kmax >> 1
%o A003158         while kmax-kmin > 1:
%o A003158             kmid = kmax+kmin>>1
%o A003158             if f(kmid) <= kmid:
%o A003158                 kmax = kmid
%o A003158             else:
%o A003158                 kmin = kmid
%o A003158         return kmax
%o A003158     def f(x):
%o A003158         c, s = n+x, bin(x)[2:]
%o A003158         l = len(s)
%o A003158         for i in range(l&1,l,2):
%o A003158             c -= int(s[i])+int('0'+s[:i],2)
%o A003158         return c
%o A003158     return bisection(f,n,n)+n-1 # _Chai Wah Wu_, Jan 29 2025
%Y A003158 Cf. A001045, A003156, A003157, A079523, A092606.
%K A003158 nonn
%O A003158 1,1
%A A003158 _N. J. A. Sloane_
%E A003158 Definition clarified by _N. J. A. Sloane_, Dec 26 2020