A328208 -id:A328208 - cp's OEIS frontend

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

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 *)

A352342 Lazy-Pell-Niven numbers: numbers that are divisible by the sum of the digits in their maximal (or lazy) representation in terms of the Pell numbers (A352339).

Original entry on oeis.org

1, 2, 4, 9, 12, 15, 20, 24, 25, 28, 30, 35, 40, 48, 50, 54, 56, 60, 63, 64, 70, 72, 78, 84, 88, 91, 96, 102, 115, 120, 136, 144, 160, 162, 168, 180, 182, 184, 189, 207, 209, 210, 216, 217, 234, 246, 256, 261, 270, 304, 306, 308, 315, 320, 328, 333, 350, 352, 357

Offset: 1

Amiram Eldar, Mar 12 2022

Numbers k such that A352340(k) | k.

			4 is a term since its maximal Pell representation, A352339(4) = 11, has the sum of digits A352340(4) = 1+1 = 2 and 4 is divisible by 2.

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

Cf. A352339, A352340.
Subsequences: A352343, A352344, A352345.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; q[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; Divisible[n, Plus @@ v[[i[[1, 1]] ;; -1]]]]; Select[Range[300], q]

A352508 Catalan-Niven numbers: numbers that are divisible by the sum of the digits in their representation in terms of the Catalan numbers (A014418).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 14, 16, 18, 21, 24, 28, 30, 32, 33, 40, 42, 44, 45, 48, 55, 56, 57, 60, 65, 72, 78, 80, 84, 88, 95, 100, 105, 112, 126, 128, 130, 132, 134, 135, 138, 140, 144, 145, 146, 147, 152, 155, 156, 168, 170, 174, 180, 184, 185, 195, 210, 216

Offset: 1

Amiram Eldar, Mar 19 2022

Numbers k such that A014420(k) | k.
All the Catalan numbers (A000108) are terms.
If k is an odd Catalan number (A038003), then k+1 is a term.

			4 is a term since its Catalan representation, A014418(4) = 20, has the sum of digits A014420(4) = 2 + 0 = 2 and 4 is divisible by 2.
9 is a term since its Catalan representation, A014418(9) = 120, has the sum of digits A014420(9) = 1 + 2 + 0 = 3 and 9 is divisible by 3.

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

Cf. A014418, A014420.
Subsequences: A000108, A038003, A352509, A352510, A352511.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342.

c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; Select[Range[216], q]

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[-#]] &]

A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 16, 20, 22, 24, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 68, 72, 75, 76, 84, 86, 87, 88, 92, 93, 95, 96, 99, 100, 104, 105, 108, 112, 115, 117, 120, 125, 126, 128, 129, 132, 135, 136, 138, 140

Offset: 1

Amiram Eldar, Jul 14 2023

Numbers k such that A364215(k) | k.

A007583 is a subsequence since A364215(A007583(n)) = 1 for n >= 0.

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

Cf. A280049, A364215.
Subsequences: A007583, A364217, A364218, A364219, A364220, A364221.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

seq[kmax_] := Module[{m = 1, s = {}}, Do[If[Divisible[k, DigitCount[m, 2, 1]], AppendTo[s, k]]; While[m++; OddQ[IntegerExponent[m, 2]]], {k, 1, kmax}]; s]; seq[140]

lista(kmax) = {my(m = 1); for(k = 1, kmax, if( !(k % sumdigits(m, 2)), print1(k,", ")); until(valuation(m, 2)%2 == 0, m++));}

A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 20, 21, 22, 24, 26, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 64, 68, 69, 72, 75, 76, 80, 84, 85, 86, 87, 88, 90, 92, 93, 96, 99, 100, 104, 105, 106, 108, 111, 112, 115, 116, 117, 120

Offset: 1

Amiram Eldar, Jul 21 2023

Numbers k such that A265745(k) | k.

The positive Jacobsthal numbers, A001045(n) for n >= 1, are terms since their representation in Jacobsthal greedy base is one 1 followed by n-1 0's, so A265745(A001045(n)) = 1 divides A001045(n).

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

Cf. A265745, A265747.
Subsequences: A364380, A364381, A364382, A364383.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508, A364216.

greedyJacobNivenQ[n_] := Divisible[n, A265745[n]]; Select[Range[120], greedyJacobNivenQ] (* using A265745[n] *)

isA364379(n) = !(n % A265745(n)); \\ using A265745(n)

A331819 Positive numbers k such that -k is a negative negabinary-Niven number, i.e., divisible by the sum of digits of its negabinary representation (A027615).

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 33, 34, 36, 39, 40, 42, 44, 48, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 77, 78, 80, 84, 90, 92, 96, 100, 102, 104, 108, 111, 112, 114, 115, 116, 120, 123, 124, 126, 128, 129, 130, 132, 135, 136, 138, 140

Offset: 1

Amiram Eldar, Jan 27 2020

			6 is a term since A039724(-6) = 1110 and 1 + 1 + 1 + 0 = 3 is a divisor of 6.

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

Cf. A005352, A027615, A039724, A005349, A049445, A328208, A328212, A331085, A331088, A331728.

negaBinWt[n_] := negaBinWt[n] = If[n==0, 0, negaBinWt[Quotient[n-1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; Select[Range[100], negaBinNivenQ]

A333619 Numbers that are divisible by the total number of 1's in the Zeckendorf representations of all their divisors (A300837).

Original entry on oeis.org

1, 2, 4, 10, 15, 18, 20, 25, 44, 55, 56, 63, 70, 78, 80, 96, 108, 126, 128, 190, 275, 324, 338, 341, 416, 442, 451, 484, 494, 517, 520, 550, 637, 682, 720, 726, 736, 760, 780, 781, 803, 816, 845, 946, 990, 1088, 1111, 1113, 1199, 1235, 1239, 1311, 1426, 1441

Offset: 1

Amiram Eldar, Mar 29 2020

			4 is a term since its divisors are {1, 2, 4}, their Zeckendorf representations (A014417) are {1, 10, 101}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) = 4 which is a divisor of 4.

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

Cf. A007895, A014417, A093705, A300837, A328208, A333617, A333620.

zeckDigSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
zeckDivDigSum[n_] := DivisorSum[n, zeckDigSum[#] &];
Select[Range[10^3], Divisible[#, zeckDivDigSum[#]] &]

A381581 Numbers divisible by the sum of the digits in their Chung-Graham representation (A381579).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 20, 21, 22, 24, 27, 28, 30, 40, 42, 44, 45, 48, 55, 56, 57, 58, 60, 66, 70, 72, 75, 76, 80, 84, 90, 92, 95, 96, 100, 102, 110, 111, 112, 115, 116, 120, 132, 135, 138, 140, 144, 150, 152, 153, 156, 168, 170, 175, 176, 180, 186, 190, 195, 198

Offset: 1

Amiram Eldar, Feb 28 2025

Numbers k such that A291711(k) divides k.
Analogous to Niven numbers (A005349) with the Chung-Graham representation (A381579) instead of the decimal representation.
A001906(k) = Fibonacci(2*k) is a term for all k >= 1.
If k is not divisible by 3 (A001651), then Fibonacci(2*k) + 1 is a term.

			4 is a term since A291711(4) = 1 divides 4.
6 is a term since A291711(6) = 2 divides 6.

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

Cf. A000045, A001651, A001906, A291711.
Subsequences: A381582, A381583, A381584, A381585.
Similar sequences: A005349, A049445, A064150, A328208, A328212.

f[n_] := f[n] = Fibonacci[2*n]; q[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; Divisible[n, s]]; Select[Range[200], q]

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
isok(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); !(n % s);}

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

cp's OEIS Frontend

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

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A352342 Lazy-Pell-Niven numbers: numbers that are divisible by the sum of the digits in their maximal (or lazy) representation in terms of the Pell numbers (A352339).

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A352508 Catalan-Niven numbers: numbers that are divisible by the sum of the digits in their representation in terms of the Catalan numbers (A014418).

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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).

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A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

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A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

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A331819 Positive numbers k such that -k is a negative negabinary-Niven number, i.e., divisible by the sum of digits of its negabinary representation (A027615).

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A333619 Numbers that are divisible by the total number of 1's in the Zeckendorf representations of all their divisors (A300837).

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A381581 Numbers divisible by the sum of the digits in their Chung-Graham representation (A381579).