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A377209 Zeckendorf-Niven numbers (A328208) k such that k/z(k) is also a Zeckendorf-Niven number, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

%I A377209 #7 Oct 20 2024 13:55:30
%S A377209 1,2,3,4,5,6,8,10,12,13,16,21,24,26,30,34,36,42,48,55,60,66,68,72,78,
%T A377209 81,89,90,108,110,120,126,135,144,152,168,178,180,192,204,207,233,240,
%U A377209 243,264,270,276,288,300,304,312,324,330,336,360,377,380,390,396,408
%N A377209 Zeckendorf-Niven numbers (A328208) k such that k/z(k) is also a Zeckendorf-Niven number, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.
%H A377209 Amiram Eldar, <a href="/A377209/b377209.txt">Table of n, a(n) for n = 1..10000</a>
%e A377209 12 is a term since 12/z(12) = 4 is an integer and also 4/z(4) = 2 is an integer.
%t A377209 zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* _Alonso del Arte_ at A007895 *)
%t A377209 q[k_] := Module[{z = zeck[k]}, Divisible[k, z] && Divisible[k/z, zeck[k/z]]]; Select[Range[400], q]
%o A377209 (PARI) zeck(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
%o A377209 is(k) = {my(z = zeck(k)); !(k % z) && !((k/z) % zeck(k/z)); }
%Y A377209 Cf. A007895, A376616 (binary analog).
%Y A377209 Subsequence of A328208.
%Y A377209 Subsequences: A000045, A377210.
%K A377209 nonn,easy,base
%O A377209 1,2
%A A377209 _Amiram Eldar_, Oct 20 2024