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A374960
Numbers k such that 2^k and 2^(k+1) have the same number of terms in their Zeckendorf representation (A007895).
2
0, 5, 6, 7, 11, 18, 20, 25, 39, 52, 61, 96, 104, 157, 176, 199, 206, 210, 279, 326, 333, 339, 369, 380, 397, 411, 426, 473, 542, 576, 743, 860, 898, 921, 961, 970, 993, 1024, 1043, 1049, 1100, 1121, 1176, 1184, 1193, 1199, 1206, 1230, 1253, 1376, 1380, 1387, 1435
OFFSET
1,2
COMMENTS
Numbers k such that A020908(k) = A020908(k+1).
The corresponding values of A020908(k) are 1, 3, 3, 3, 6, 7, 8, 9, 18, 20, 28, 44, 37, ... .
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..550 (terms below 10^5)
EXAMPLE
0 is a term since the Zeckendorf representation of 2^0 = 1 is A014417(1) = 1, and the Zeckendorf representation of 2^1 = 2 is A014417(2) = 10, so A020908(0) = A020908(1) = 1.
5 is a term since the Zeckendorf representation of 2^5 = 32 is A014417(32) = 1010100, and the Zeckendorf representation of 2^6 = 64 is A014417(64) = 100010001, so A020908(5) = A020908(6) = 3.
MATHEMATICA
z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
s[n_] := s[n] = z[2^n]; Select[Range[0, 1500], s[#] == s[# + 1] &]
PROG
(PARI) A007895(n)=if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s); \\ Charles R Greathouse IV at A007895
lista(kmax) = {my(z1 = A007895(1), z2); for(k = 1, kmax, z2 = A007895(2^k); if(z1 == z2, print1(k-1 , ", ")); z1 = z2); }
CROSSREFS
A374961 is a subsequence.
Sequence in context: A064615 A300957 A139205 * A288859 A307751 A039589
KEYWORD
nonn,base
AUTHOR
Amiram Eldar, Jul 25 2024
STATUS
approved