A055778 -id:A055778 - cp's OEIS frontend

A334308 Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation (A055778).

1, 2, 6, 12, 15, 16, 18, 20, 30, 35, 36, 45, 48, 55, 60, 70, 72, 78, 84, 90, 91, 95, 96, 98, 104, 108, 132, 144, 147, 154, 168, 175, 184, 189, 208, 224, 231, 232, 245, 252, 256, 261, 264, 270, 275, 280, 282, 287, 294, 315, 322, 324, 330, 336, 340, 342, 351, 357

Offset: 1

Amiram Eldar, Apr 22 2020

			6 is a term since its base phi representation is 1010.0001, and the number of 1's is 3, which is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
Wikipedia, Golden ratio base.

Cf. A005349, A049445, A055778, A064150, A064438, A064481, A118363, A328208, A328212, A333426.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n]] ][[1]]; Select[Range[360], Divisible[#, phiDigSum[#]] &]

A339213 Phi-base self numbers: positive numbers not of the form k + A055778(k).

Original entry on oeis.org

1, 3, 6, 10, 12, 15, 19, 23, 26, 30, 32, 38, 41, 43, 52, 55, 59, 61, 64, 68, 72, 75, 79, 81, 86, 89, 91, 97, 101, 104, 108, 110, 115, 118, 120, 126, 131, 135, 137, 140, 144, 148, 151, 155, 157, 163, 166, 168, 177, 180, 184, 186, 189, 193, 197, 200, 204, 206, 213

Offset: 1

Amiram Eldar, Nov 27 2020

Analogous to self numbers (A003052) using base phi (A130600) instead of base 10.

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
F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Golden ratio base.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A010061, A010064, A010067, A010070, A055778, A130600, A331083, A334308, A339211, A339212, A339214, A339215.

s[1] = 2; s[n_] := n + Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]][[1]]; m = 220; Complement[Range[m], Array[s, m]]

A190720 Numbers k such that A055778(k) < A055778(k-1).

Original entry on oeis.org

7, 12, 16, 18, 25, 30, 34, 36, 41, 45, 47, 54, 59, 63, 65, 72, 77, 81, 83, 88, 92, 94, 101, 106, 110, 112, 117, 121, 123, 130, 135, 139, 141, 148, 153, 157, 159, 164, 168, 170, 177, 182, 186, 188, 195, 200, 204, 206, 211, 215, 217, 224, 229, 233

Offset: 1

Carmine Suriano, May 17 2011

nonn

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

Cf. A055778, A190721.

A190721 Numbers k such that A055778(k) = A055778(k-1).

Original entry on oeis.org

3, 5, 6, 10, 13, 14, 17, 21, 23, 24, 28, 31, 32, 35, 39, 42, 43, 46, 50, 52, 53, 57, 60, 61, 64, 68, 70, 71, 75, 78, 79, 82, 86, 89, 90, 93, 97, 99, 100, 104, 107, 108, 111, 115, 118, 119, 122, 126, 128, 129, 133, 136, 137, 140, 144, 146, 147, 151

Offset: 1

Carmine Suriano, May 17 2011

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

Cf. A055778, A190720.

A190723 Numbers m for which A055778(m) > A055778(m-1).

Original entry on oeis.org

1, 2, 4, 8, 9, 11, 15, 19, 20, 22, 26, 27, 29, 33, 37, 38, 40, 44, 48, 49, 51, 55, 56, 58, 62, 66, 67, 69, 73, 74, 76, 80, 84, 85, 87, 91, 95, 96, 98, 102, 103, 105, 109, 113, 114, 116, 120, 124, 125, 127, 131, 132, 134, 138, 142, 143, 145, 149, 150, 152, 156

Offset: 1

Carmine Suriano, May 17 2011

nonn

The sequence (a(n+1)-1) = 1,3,7,8,10,... is the union of two generalized Beatty sequences, namely (floor(n*phi)+2*n) = A003231, and the sequence (4*floor(n*phi)+3*n+1), the latter with offset 0. For a proof see my paper "Points of increase...". - Michel Dekking, Apr 01 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michel Dekking, Points of increase of the sum of digits function of the base phi expansion, arXiv:2003.14125 [math.CO], 2020.

Cf. A055778, A190720, A190721, A003231. The morphism in the Formula is a change of alphabet of the morphism generating A284749.

Block[{nn = 160, k}, k = 2 Ceiling[Log[GoldenRatio, nn]]; Position[Differences@ Array[Total@ First@ RealDigits[#, GoldenRatio, k] &, nn, 0], ?(# > 0 &)][[All, 1]]] (* _Michael De Vlieger, Apr 02 2020, after T. D. Noe at A055778 *)

a(n) = 1 + Sum_{k=1..n-1} x(k), where x is the unique fixed point of the morphism 1->12, 2->4, 4->1244 on the alphabet {1,2,4}. - Michel Dekking, Apr 01 2020

A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

Original entry on oeis.org

0, 1, 10, 100, 101, 1000, 1010, 10000, 10001, 10010, 10100, 10101, 100000, 100010, 100100, 100101, 101000, 101010, 1000000, 1000001, 1000010, 1000100, 1000101, 1001000, 1001010, 1010000, 1010001, 1010010, 1010100, 1010101, 10000000

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 08 2005

			2 = 10.01 in base phi, so left of the decimal point is 10.
The first few numbers written in base phi:
0 = 0.
1 = 1.
2 = 10.01
3 = 100.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 10000.0001
8 = 10001.0001
9 = 10010.0101
10 = 10100.0101
11 = 10101.0101
12 = 100000.101001
13 = 100010.001001
14 = 100100.001001
15 = 100101.001001
16 = 101000.100001
17 = 101010.000001
18 = 1000000.000001
19 = 1000001.000001
20 = 1000010.010001
21 = 1000100.010001
22 = 1000101.010001
23 = 1001000.100101
24 = 1001010.000101
...

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Michel Dekking, How to add two natural numbers in base phi, arXiv:2002.01665 [math.NT], 5 Feb 2020.
Ron Knott, Phigits and the Base Phi representation.
Ron Knott, Phigits and the Base Phi representation [Local copy, pdf only]
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. [Note that this document has been revised multiple times.]

Cf. A001622, A055778, A105425, A104605.
See A341722 for the part to the right of the decimal point.
Cf. A105116 (base e), A344939 (base Pi).

nn = 1000; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; FromDigits[Take[d[[1]], d[[2]]]], {n, 0, nn}] (* T. D. Noe, May 20 2011 *)

Definition clarified by N. J. A. Sloane, May 27 2023

A130600 Integers written in base phi, with the "decimal point" omitted.

Original entry on oeis.org

1, 1001, 10001, 10101, 10001001, 10100001, 100000001, 100010001, 100100101, 101000101, 101010101, 100000101001, 100010001001, 100100001001, 100101001001, 101000100001, 101010000001, 1000000000001, 1000001000001

Offset: 1

Casey Mongoven, Aug 06 2007

This is the "greedy" or "minimal" representation (see also A130601).

			If the decimal point were included, the sequence would read 1., 10.01, 100.01, 101.01, 1000.1001, 1010.0001, 10000.0001, 10001.0001, 10010.0101, 10100.0101, 10101.0101, ... Unfortunately these are not integers.
Examples: a(2)=1001 because phi^1+phi^-2 = 2, a(3) = 10001 because phi^2+phi^-2 = 3, a(4) = 10101 because phi^2+phi^0+phi^-2 = 4.

Casey Mongoven and T. D. Noe, Table of n, a(n) for n = 1..1000
Casey Mongoven, Music based on this sequence
Ron Knott, Integers written in base phi

Cf. A001622, A055778, A105424, A130601.

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; last1 = Position[d[[1]], 1][[-1, 1]]; FromDigits[Take[d[[1]], last1]], {n, nn}] (* T. D. Noe, May 20 2011 *)

A214971 Integers k for which the base-phi representation of k includes 1.

Original entry on oeis.org

1, 4, 8, 11, 15, 19, 22, 26, 29, 33, 37, 40, 44, 48, 51, 55, 58, 62, 66, 69, 73, 76, 80, 84, 87, 91, 95, 98, 102, 105, 109, 113, 116, 120, 124, 127, 131, 134, 138, 142, 145, 149, 152, 156, 160, 163, 167, 171, 174, 178, 181, 185, 189, 192, 196, 199, 203

Offset: 1

Clark Kimberling, Oct 17 2012

Conjecture:  L(2k-1) and L(2k)+1 are terms of this sequence for all positive integers k, where L=A000032 (Lucas numbers).
Proof of this conjecture: this follows directly from the well known formula L(2k)=phi^{2k}+phi^{-2k}, and the recursion L(2k+1)=L(2k)+L(2k-1). - Michel Dekking, Jun 25 2019
Conjecture: If D is the difference sequence, then D-3 is the infinite Fibonacci word A096270. If so, then A214971 can be generated as in Program 3 of the Mathematica section. - Peter J. C. Moses, Oct 19 2012
Conjecture: A very simple formula for this sequence seems to be a(n) = ceiling((n-1)*phi) + 2*(n-1) for n>1; thus, see the related sequence A004956. - Thomas Baruchel, May 14 2018
Moses' conjecture is equivalent to Baruchel's conjecture: Baruchel's conjecture expresses that this sequence is a generalized Beatty sequence, and since A096270 equals the Fibonacci word A005614 with an initial zero, this follows directly from Lemma 8 in Allouche and Dekking. - Michel Dekking, May 04 2019
The conjectures by Baruchel and Moses are proved in my paper 'Base phi representations and golden mean beta-expansions'. - Michel Dekking, Jun 25 2019
a(n) equals A198270(n-1) for 0A198270(n-1) or A198270(n-1)+1 for all n<90, after which the two sequences very slowly diverge from each other. - Greg Dresden, Aug 15 2020

			1 = 1,
4 = r^2 + 1 + 1/r^2,
8 = r^4 + 1 + 1/r^4,
11 = r^4 + r^1 + 1 + 1/r^2 + 1/r^4.
where r = phi = (1 + sqrt(5))/2 = the golden ratio.

Clark Kimberling, Table of n, a(n) for n = 1..2000
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
Michel Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.

Cf. A055778, A214969, A214970, A000032, A096270.

(* 1st program *)
r = GoldenRatio; f[x_] := Floor[Log[r, x]];
t[n_] := RealDigits[n, r, 1000]
p[n_] := Flatten[Position[t[n][[1]], 1]]
Table[{n, f[n] + 1 - p[n]}, {n, 1, 47}] (* {n, exponents of r in base phi repr of n} *)
m[n_] := If[MemberQ[f[n] + 1 - p[n], 0], 1, 0]
u = Table[m[n], {n, 1, 900}]
Flatten[Position[u, 1]]  (* A214971 *)
(* 2nd program *)
A214971 = Map[#[[1]] &, Cases[Table[{n, Last[#] - Flatten[Position[First[#], 1]] &[RealDigits[n, GoldenRatio, 1000]]}, {n, 1, 5000}], {, {__, 0, _}}]] (* Peter J. C. Moses, Oct 19 2012 *)
(* 3rd program; see Comments *)
Accumulate[Flatten[{1, Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 1, 1}}] &, {0}, 8] + 3}]]  (* Peter J. C. Moses, Oct 19 2012 *)

from math import isqrt
def A214971(n): return (n<<1)-1+(n-1+isqrt(5*(n-1)**2)>>1) # Chai Wah Wu, May 22 2025

a(n) = floor((n-1)*phi) + 2*n - 1. - Primoz Pirnat, Jun 09 2024

A178482 Phi-antipalindromic numbers.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 11, 18, 19, 21, 22, 25, 26, 28, 29, 47, 48, 50, 51, 54, 55, 57, 58, 65, 66, 68, 69, 72, 73, 75, 76, 123, 124, 126, 127, 130, 131, 133, 134, 141, 142, 144, 145, 148, 149, 151, 152, 170, 171, 173, 174

Offset: 1

Vladimir Shevelev, May 28 2010

We call m a phi-antipalindromic number if for the vector (a,...,b) (a<...=2, either a(n)+1 or a(n)-1 is in the sequence; also either a(n)+3 or a(n)-3 is in the sequence.
Conjecture: this is the sequence of numbers k for which f(k) is an integer, where f(x) is the change-of-base function defined at A214969 using b=phi and c=b^2. - Clark Kimberling, Oct 17 2012
There is a 21-state automaton accepting the Zeckendorf representations of those n in this sequence. - Jeffrey Shallit, May 03 2023
Kimberling's conjecture has been proven by Ingrid Vukusic and myself. Along the way we prove an alternate characterization of the sequence:  they are the positive integers whose base-phi expansion consists only of even exponents of phi. - Jeffrey Shallit, Aug 28 2025
Alternatively, this sequence consists of those numbers k such that either k or k-1 can be written as the (possibly empty) sum of distinct Lucas numbers L_i where i>=2 and i is even. - Jeffrey Shallit, Aug 28 2025

			The vectors of exponents of 4 and 5 are (-2,0,2) and (-4,-1,3) correspondingly (cf.A104605). Therefore by definition 4 is a phi-antipalindromic number, while 5 is not. Let n=38. Then k=5. Thus a(38)=A005248(5)+a(6)=123+10=133. The vector of exponents of phi in the base-phi expansion of 133 is (-10,-4,-2,2,4,10).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3071 from R. J. Mathar)
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A005248, A055778, A104605, A104626, A104627, A104628.

For bisections see A171070, A171071.

phiAPQ[1] = True; phiAPQ[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]] - Flatten @ Position[d[[1]], 1]; Reverse[e] == -e]; Select[Range[200], phiAPQ] (* Amiram Eldar, Apr 23 2020 *)

For k>=1, a(2^k)=A005248(k); if 2^k

A104605 Triangle read by rows: row n gives list of powers of phi in the representation of the integer n as a sum of increasing nonconsecutive powers of the golden ratio.

Original entry on oeis.org

0, -2, 1, -2, 2, -2, 0, 2, -4, -1, 3, -4, 1, 3, -4, 4, -4, 0, 4, -4, -2, 1, 4, -4, -2, 2, 4, -4, -2, 0, 2, 4, -6, -3, -1, 5, -6, -3, 1, 5, -6, -3, 2, 5, -6, -3, 0, 2, 5, -6, -1, 3, 5, -6, 1, 3, 5, -6, 6, -6, 0, 6, -6, -2, 1, 6, -6, -2, 2, 6, -6, -2, 0, 2, 6, -6, -4, -1, 3, 6, -6, -4, 1, 3, 6, -6, -4, 4, 6, -6, -4, 0, 4, 6, -6, -4, -2, 1, 4, 6, -6, -4

Offset: 1

Eric W. Weisstein, Mar 17 2005

Let f(n) = F(n+1) = A000045(n) and extend n to include negative indices. Then each row n can equally well be thought of as a sequence a_1, a_2,..., a_k such that f(a_1) + f(a_2) + ... + f(a_k) = n. For example, the fifth row is -4 -1 3, so f(-4) + f(-1) + f(3) = 2 + 0 + 3 = 5. - Dale Gerdemann, Apr 01 2012

			   0
  -2  1
  -2  2
  -2  0  2
  -4 -1  3
  -4  1  3
  -4  4
  -4  0  4
  ...
phi^0, phi^(-2) + phi, phi^(-2) + phi^2, phi^(-2) + phi^0 + phi^2, ...

T. D. Noe, Rows n = 1..1000, flattened
Dale Gerdemann, Combinatorial proofs of Zeckendorf family identities, Fib. Quart. 46/47 (2009) 249.
Eric Weisstein's World of Mathematics, Phi Number System

Cf. A055778 (length of row n), A105424, A178482 (phi-antipalindromic numbers).

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; Reverse[d[[2]] - Flatten[Position[d[[1]], 1]]], {n, nn}] (* T. D. Noe, May 20 2011 *)

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cp's OEIS Frontend

A334308 Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation (A055778).

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A339213 Phi-base self numbers: positive numbers not of the form k + A055778(k).

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A190720 Numbers k such that A055778(k) < A055778(k-1).

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A190721 Numbers k such that A055778(k) = A055778(k-1).

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A190723 Numbers m for which A055778(m) > A055778(m-1).

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A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

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A130600 Integers written in base phi, with the "decimal point" omitted.

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A214971 Integers k for which the base-phi representation of k includes 1.

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A178482 Phi-antipalindromic numbers.

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