A364121 -id:A364121 - cp's OEIS frontend

A364123 Stolarsky-Niven numbers: numbers that are divisible by the number of 1's in their Stolarsky representation (A364121).

2, 4, 6, 8, 9, 12, 14, 16, 20, 22, 24, 27, 30, 36, 38, 40, 42, 44, 48, 54, 56, 57, 60, 65, 69, 72, 75, 80, 84, 85, 90, 92, 96, 98, 100, 102, 104, 108, 112, 116, 120, 124, 126, 132, 136, 138, 145, 147, 150, 153, 155, 159, 160, 175, 180, 185, 190, 195, 196, 205

Offset: 1

Amiram Eldar, Jul 07 2023

Numbers k such that A200649(k) | k.

Fibonacci(k) + 1 is a term if k !== 3 (mod 6) (i.e., k is in A047263).

			4 is a term since its Stolarsky representation, A364121(4) = 10, has one 1 and 4 is divisible by 1.
6 is a term since its Stolarsky representation, A364121(6) = 101, has 2 1's and 6 is divisible by 2.

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

Cf. A047263, A200649, A364121.
Subsequences: A364124, A364125, A364126.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
stolNivQ[n_] := n > 1 && Divisible[n, Total[stol[n]]];
Select[Range[200], stolNivQ]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
isA364123(n) = n > 1 && !(n % vecsum(stol(n)));

A364122 Numbers whose Stolarsky representation (A364121) is palindromic.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 13, 15, 18, 21, 23, 34, 36, 40, 45, 50, 55, 66, 71, 89, 91, 95, 108, 113, 120, 128, 136, 144, 159, 176, 196, 204, 233, 235, 239, 261, 273, 286, 291, 298, 319, 327, 338, 351, 364, 377, 400, 426, 464, 490, 518, 550, 563, 610, 612, 616, 654, 667

Offset: 1

Amiram Eldar, Jul 07 2023

The positive Fibonacci numbers (A000045) are terms since the Stolarsky representation of Fibonacci(1) = Fibonacci(2) is 0 and the Stolarsky representation of Fibonacci(n) is n-2 1's for n >= 3.

Fiboancci(2*n+1) + 2 is a term for n >= 3, since its Stolarsky representation is n-1 0's between two 1's.

			The first 10 terms are:
   n  a(n)  A364121(a(n))
  --  ----  -------------
   1     1  0
   2     2  1
   3     3  11
   4     5  111
   5     6  101
   6     8  1111
   7    13  11111
   8    15  1001
   9    18  11011
  10    21  111111

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

Cf. A000045, A200648, A200649, A200650, A200651, A200714, A364121.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
stolPalQ[n_]:= PalindromeQ[stol[n]]; Select[Range[700], stolPalQ]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
is(n) = {my(s = stol(n)); s == Vecrev(s);}

A364127 The number of trailing 0's in the Stolarsky representation of n (A364121).

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Jul 07 2023

The first position of k = 2, 3, 4, ... is A055588(k+1).
The asymptotic density of the occurrences of k = 0, 1, 2, ... is (2-phi)^k/phi, where phi is the golden ratio (A001622).
The asymptotic mean of this sequence is phi - 1 (A094214) and the asymptotic standard deviation is 1.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A001622, A055588, A094214, A122840, A364121.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
a[n_] := IntegerExponent[FromDigits[stol[n]], 10]; Array[a, 100, 2]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
a(n) = valuation(fromdigits(stol(n)), 10);

a(n) = A122840(A364121(n)).

cp's OEIS Frontend

A364123 Stolarsky-Niven numbers: numbers that are divisible by the number of 1's in their Stolarsky representation (A364121).

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A364122 Numbers whose Stolarsky representation (A364121) is palindromic.

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A364127 The number of trailing 0's in the Stolarsky representation of n (A364121).

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