A007895 -id:A007895 - cp's OEIS frontend

A328208 Zeckendorf-Niven numbers: numbers divisible by the number of terms in their Zeckendorf representation (A007895).

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 16, 18, 21, 22, 24, 26, 27, 30, 34, 36, 42, 45, 48, 55, 56, 58, 60, 66, 68, 69, 72, 76, 78, 80, 81, 84, 89, 90, 92, 93, 94, 96, 99, 102, 105, 108, 110, 111, 116, 120, 126, 132, 135, 140, 144, 146, 150, 152, 153, 156, 159, 162

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			12 is in the sequence since A007895(12) = 3 and 3 is a divisor of 12.

Andrew Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.

Robert Israel, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.
Andrew Ray and Curtis Cooper, On the natural density of the k-Zeckendorf Niven numbers, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

Cf. A005349, A007895.

fib:= combinat:-fibonacci:
phi:= 1/2 + sqrt(5)/2:
fibapp:= n -> phi^n/sqrt(5):
invfib := proc(x::posint)
  local q, n;
  q:= evalf((ln(x+1/2) + ln(5)/2)/ln(phi));
  n:= floor(q);
  if fib(n) <= x then
    while fib(n+1) <= x do
      n := n+1
    end do
  else
    while fib(n) > x do
      n := n-1
    end do
  end if;
  n
end proc:
zeck:= proc(x) local n;
 if x = 0 then 0
 else
   n:= invfib(x);
   F[n] + zeck(x-fib(n));
 fi
end proc:
filter:= n -> n mod nops(zeck(n)) = 0:
select(filter, [$1..200]); # Robert Israel, Oct 25 2019

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; Select[Range[1000], aQ] (* after Alonso del Arte at A007895 *)

A339211 Zeckendorf self numbers: numbers not of the form k + A007895(k).

Original entry on oeis.org

1, 5, 7, 10, 19, 21, 27, 29, 32, 36, 40, 42, 45, 54, 61, 63, 66, 75, 77, 83, 85, 88, 95, 97, 100, 109, 111, 117, 119, 122, 126, 130, 132, 135, 144, 146, 150, 152, 155, 164, 166, 172, 174, 177, 181, 185, 187, 190, 199, 206, 208, 211, 220, 222, 228, 230, 233, 239

Offset: 1

Amiram Eldar, Nov 27 2020

Analogous to self numbers (A003052) using the Zeckendorf representation (A014417) instead of decimal expansion.

The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 4, 25, 236, 2351, 23495, 234949, 2349463, 23494586, 234945839, 2349458364, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23494583... . - Amiram Eldar, Aug 08 2025

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Self Number.
Eric Weisstein's World of Mathematics, Zeckendorf Representation.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A007895, A010061, A010064, A010067, A010070, A014417, A328208, A339212, A339213, A339214, A339215.

z[n_] := n + Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; m = 250; Complement[Range[m], Array[z, m]] (* after Alonso del Arte at A007895 *)

A318464 Additive with a(p^e) = A007895(e), where A007895(n) gives the number of terms in Zeckendorf representation of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

Offset: 1

Antti Karttunen, Aug 30 2018

nonn

From Amiram Eldar, Aug 09 2024: (Start)
The number of factors of n of the form p^Fibonacci(k), where p is a prime and k >= 2, when the factorization is uniquely done using the Zeckendorf representation of the exponents in the prime factorization of n.
Equivalently, the number of Zeckendorf-infinitary divisors of n (defined in A318465) that are prime powers (A246655). (End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A007895, A077761, A246655, A318465, A318469.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; a[n_] := Total[z /@ FactorInteger[n][[;; , 2]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 15 2023 *)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A318464(n) = vecsum(apply(e -> A007895(e),factor(n)[,2]));

a(n) = A007814(A318465(n)).
a(n) = A001222(A318469(n)).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{k>=2} (A007895(k)-A007895(k-1)) * P(k) = 0.05631817952062180045..., where P(s) is the prime zeta function. - Amiram Eldar, Oct 09 2023

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.

Original entry on oeis.org

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, 81, 89, 90, 108, 110, 120, 126, 135, 144, 152, 168, 178, 180, 192, 204, 207, 233, 240, 243, 264, 270, 276, 288, 300, 304, 312, 324, 330, 336, 360, 377, 380, 390, 396, 408

Offset: 1

Amiram Eldar, Oct 20 2024

			12 is a term since 12/z(12) = 4 is an integer and also 4/z(4) = 2 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007895, A376616 (binary analog).
Subsequence of A328208.
Subsequences: A000045, A377210.

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
q[k_] := Module[{z = zeck[k]}, Divisible[k, z] && Divisible[k/z, zeck[k/z]]]; Select[Range[400], q]

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
is(k) = {my(z = zeck(k)); !(k % z) && !((k/z) % zeck(k/z)); }

A377210 Zeckendorf-Niven numbers (A328208) k such that m = k/z(k) and m/z(m) are also Zeckendorf-Niven numbers, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 21, 24, 26, 30, 34, 42, 48, 55, 60, 68, 78, 89, 110, 120, 126, 144, 178, 180, 192, 204, 233, 243, 264, 270, 288, 300, 312, 324, 330, 360, 377, 466, 480, 534, 540, 576, 600, 610, 621, 672, 720, 754, 768, 864, 987, 1020, 1056

Offset: 1

Amiram Eldar, Oct 20 2024

			24 is a term since 24/z(24) = 12, 12/z(12) = 4 and 4/z(4) = 2 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045 (a subsequence), A007895, A376617 (binary analog).

Subsequence of A328208 and A377209.

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
q[k_] := Module[{z = zeck[k], z2, m, n}, IntegerQ[m = k/z] && Divisible[m, z2 = zeck[m]] && Divisible[n = m/z2, zeck[n]]]; Select[Range[1000], q]

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
is(k) = {my(z = zeck(k), z2, m); if(k % z, return(0)); m = k/z; z2 = zeck(m); !(m % z2) && !((m/z2) % zeck(m/z2)); }

A220115 a(n) = A000120(n) - A007895(n), the number of 1's in binary expansion of n minus the number of terms in Zeckendorf representation of n.

Original entry on oeis.org

0, 0, 0, 1, -1, 1, 0, 1, 0, 0, 0, 1, -1, 2, 1, 2, -1, -1, 0, 0, -1, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, -2, -2, 1, 1, 0, 1, 0, 2, -1, 0, 1, 1, 0, 1, 0, 3, -1, 0, 0, 0, 0, 0, 0, 4, 1, 2, 2, 2, 2, 2, 2, 4, -2, -1, -1, -1, 0, 0, 0, 1, -2, 0, -1, 0, 1, 1, 1, 2, -2

Offset: 0

Alex Ratushnyak, Dec 05 2012

			a(4) = A000120(4) - A007895(4) = 1 - 2 = -1.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000120, A007895.

zeck = DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1]; DigitCount[Range[0, Length[zeck]-1], 2, 1] - zeck (* Jean-François Alcover, Jan 25 2018 *)

a(n) = A000120(n) - A007895(n).

A324905 a(n) = A007895(A003965(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 2, 3, 2, 3, 3, 1, 3, 1, 3, 2, 2, 3, 2, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 3, 3, 2, 3, 1, 4, 3, 3, 2, 2, 1, 2, 1, 2, 4, 3, 3, 3, 1, 3, 2, 4, 1, 4, 1, 2, 4, 3, 3, 3, 1, 4, 3, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 1, 3, 4, 3, 1, 3, 1, 3, 4

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17711
Index entries for sequences computed from indices in prime factorization

Cf. A003965, A007895, A300837, A324888, A324907.

A003965(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = fibonacci(2+primepi(f[i, 1]))); factorback(f); };
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A324905(n) = A007895(A003965(n));

a(n) = A007895(A003965(n)).

A324907 a(n) = A007895(A113175(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 4, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 4, 1, 2, 1, 3, 1, 1, 4, 3, 3, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17711

Cf. A113175, A007895, A300837, A324888, A324905.

A113175(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = fibonacci(f[i, 1])); factorback(f); };
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A324907(n) = A007895(A113175(n));

a(n) = A007895(A113175(n)).

a(2n) = a(n).

A358978 Numbers that are coprime to the number of terms in their Zeckendorf representation (A007895).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 15, 17, 19, 20, 21, 23, 25, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 67, 70, 71, 73, 75, 77, 79, 83, 85, 87, 88, 89, 91, 95, 97, 98, 100, 101, 103, 104, 107, 109

Offset: 1

Amiram Eldar, Dec 07 2022

First differs from A063743 at n = 22.
Numbers k such that gcd(k, A007895(k)) = 1.
The Fibonacci numbers (A000045) are terms. These are also the only Zeckendorf-Niven numbers (A328208) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 61, 614, 6028, 61226, 606367, 6041106, 61235023, 612542436, 6034626175, 60093287082, 609082612171, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

			3 is a term since A007895(3) = 1, and gcd(3, 1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007895, A059956, A063743, A328208.
Subsequences: A000040, A000045.
Similar sequences: A094387, A339076, A358975, A358976, A358977.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; Select[Range[120], CoprimeQ[#, z[#]] &] (* after Alonso del Arte at A007895 *)

is(n) = if(n<4, 1, my(k=2, m=n, s, t); while(fibonacci(k++)<=m, ); while(k && m, t=fibonacci(k); if(t<=m, m-=t; s++); k--); gcd(n, s)==1); \\ after Charles R Greathouse IV at A007895

A179180 Partial sums of A007895.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 11, 13, 15, 17, 20, 21, 23, 25, 27, 30, 32, 35, 38, 39, 41, 43, 45, 48, 50, 53, 56, 58, 61, 64, 67, 71, 72, 74, 76, 78, 81, 83, 86, 89, 91, 94, 97, 100, 104, 106, 109, 112, 115, 119, 122, 126, 130, 131, 133, 135, 137, 140, 142, 145

Offset: 0

Walt Rorie-Baety, Jun 30 2010

nonn

Total number of summands in Zeckendorf representations of all the numbers 1,2,...,n (for n>0); see the conjecture at A214979. - Clark Kimberling, Oct 23 2012

			For n = 6, a(n) = 1+1+1+2+1+2 = 8.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Christian Ballot, On Zeckendorf and Base b Digit Sums, Fibonacci Quarterly, Vol. 51, No. 4 (2013), pp. 319-325.
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A001622, A007895, A214979.

s = Reverse[Table[Fibonacci[n + 1], {n, 1, 70}]];
t2 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2, 1]], # > 0 &]] &, Range[z]]; v[n_] := Sum[t2[[k]], {k, 1, n}];
v1 = Table[v[n], {n, 1, z}]
(* Peter J. C. Moses, Oct 18 2012 *)
DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1] // Accumulate (* Jean-François Alcover, Jan 25 2018 *)

a(n) ~ c * n * log(n), where c = (phi-1)/(sqrt(5)*log(phi)) = 0.574369... and phi is the golden ratio (A001622) (Ballot, 2013). - Amiram Eldar, Dec 09 2021

Corrected term a(17); the working list of the terms were not in order. Walt Rorie-Baety, Jun 30 2010

Extended by Clark Kimberling, Oct 23 2012

cp's OEIS Frontend

A328208 Zeckendorf-Niven numbers: numbers divisible by the number of terms in their Zeckendorf representation (A007895).

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A339211 Zeckendorf self numbers: numbers not of the form k + A007895(k).

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A318464 Additive with a(p^e) = A007895(e), where A007895(n) gives the number of terms in Zeckendorf representation of n.

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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.

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A377210 Zeckendorf-Niven numbers (A328208) k such that m = k/z(k) and m/z(m) are also Zeckendorf-Niven numbers, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

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A220115 a(n) = A000120(n) - A007895(n), the number of 1's in binary expansion of n minus the number of terms in Zeckendorf representation of n.

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A324905 a(n) = A007895(A003965(n)).

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A324907 a(n) = A007895(A113175(n)).

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