A331083 -id:A331083 - cp's OEIS frontend

A331085 Positive negaFibonacci-Niven numbers: positive numbers divisible by the number of terms in their negaFibonacci representation (A331083).

1, 2, 4, 5, 6, 9, 10, 12, 13, 14, 18, 24, 26, 27, 30, 34, 36, 48, 55, 60, 64, 68, 69, 72, 78, 84, 86, 87, 88, 89, 90, 93, 94, 96, 99, 100, 102, 108, 110, 112, 116, 120, 140, 144, 150, 155, 156, 160, 172, 176, 177, 178, 180, 183, 184, 188, 192, 195, 196, 200, 204

Offset: 1

Amiram Eldar, Jan 08 2020

The k-th Fibonacci number is a term for all odd k, since its negaFibonacci representation is 1 followed by (k-1) zeros.

			4 is a term since the negaFibonacci representation of 4 is 10010 whose sum of digits is 1 + 0 + 0 + 1 + 0 = 2 which is a divisor of 4.

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

Cf. A005349, A215022, A328208, A328212, A331083.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]]; f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i]; negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s]; Select[Range[200], Divisible[#, negaFibTermsNum[#]] &]

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]]

A331088 Positive numbers k such that -k is a negative negaFibonacci-Niven number, i.e., divisible by the number of terms in its negaFibonacci representation (A331084).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 18, 20, 21, 22, 24, 27, 30, 36, 42, 44, 45, 48, 50, 51, 54, 55, 56, 57, 58, 60, 66, 72, 75, 76, 80, 84, 90, 92, 96, 100, 104, 105, 108, 110, 111, 112, 115, 116, 120, 124, 126, 128, 129, 132, 136, 138, 141, 142, 144, 150, 152, 153, 156, 168, 170, 172, 175, 176, 180, 184, 186, 190, 192, 196, 198

Offset: 1

Amiram Eldar, Jan 08 2020

The k-th Fibonacci number is a term for all even k, since its negaFibonacci representation is 1 followed by (k-1) zeros.

			4 is a term since the negaFibonacci representation of -4 is 1010 whose sum of digits is 1 + 0 + 1 + 0 = 2 which is a divisor of 4.

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

Cf. A005349, A215023, A328208, A328212, A331083, A331084, A331085.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]];
f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i];
negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s];
Select[Range[200], Divisible[#, negaFibTermsNum[-#]] &]

A331084 The number of terms in the negaFibonacci representation of -n (A215023).

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 1, 2, 3, 2, 3, 4, 3, 3, 2, 3, 3, 2, 2, 1, 2, 3, 2, 3, 4, 3, 3, 2, 3, 4, 3, 4, 5, 4, 4, 3, 4, 4, 3, 3, 2, 3, 4, 3, 4, 4, 3, 3, 2, 3, 3, 2, 2, 1, 2, 3, 2, 3, 4, 3, 3, 2, 3, 4, 3, 4, 5, 4, 4, 3, 4, 4, 3, 3, 2, 3, 4, 3, 4, 5, 4, 4, 3, 4, 5, 4

Offset: 1

Amiram Eldar, Jan 08 2020

The Fibonacci numbers F(2*n - 1) are the indices of records of this sequence.

			The negaFibonacci representation of 2 is A215023(2) = 1001, thus a(2) = 1 + 0 + 0 + 1 = 2.

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

Cf. A000045, A215023, A331083.

Cf. A000120, A007895, A007953, A053735, A095791.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]]; f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i]; nf[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s]; a[n_] := nf[-n]; Array[a, 100]

a(A000045(2*n)) = 1.

a(A000045(2*n - 1)) = n.

cp's OEIS Frontend

A331085 Positive negaFibonacci-Niven numbers: positive numbers divisible by the number of terms in their negaFibonacci representation (A331083).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A331088 Positive numbers k such that -k is a negative negaFibonacci-Niven number, i.e., divisible by the number of terms in its negaFibonacci representation (A331084).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A331084 The number of terms in the negaFibonacci representation of -n (A215023).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula