Eugen Ionascu - cp's OEIS frontend

User: Eugen Ionascu

Eugen Ionascu has authored 2 sequences.

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)(1+i) + c(2)(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.

0, 1, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 30, 31, 34, 35, 36, 37, 40, 41, 86, 87, 90, 91, 92, 93, 96, 97, 102, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 120, 121, 126, 127, 130, 131, 132, 133, 136, 137, 150, 151

Offset: 1

Eugen Ionascu, Nov 15 2022

nonn

			6 is in the sequence since 6 = T^2 + T^3 + T^4 + T^5 + T^8, where T=1+i.

Problem 12335, American Mathematical Monthly, Vol. 129, issue 7, August-September 2022.

Wikipedia, Complex-base system.

Cf. A290884.

cpol[a_, b_] :=
Module[{u, uu, v, vv, p, pp, q, L, x, k, W}, u = a; v = b; p = {};
  W = {-2 - I, 0, -1 + 2*I};
  While[(u + 1)^2 + v^2 > 1,
   If[Mod[u + v, 2] ==
     0, {uu = (u + v)/2; vv = (v - u)/2; p = Prepend[p, 0]};,
    {uu = (u - 1 + v)/2; vv = (v + 1 - u)/2; p = Prepend[p, 1]}
    ]; u = uu; v = vv;
   ]; w = u + v*I; q = MemberQ[W, w]; L = Length[p];
  If[q == True, pp[x_] := Sum[p[[k]]*x^(L - k), {k, 1, L}],
   pp[x_] := "No writing"] ; {w, pp[1 + I], pp[T]}]

is(n)= while (n, if (n==I, return (0), real(n)%2==imag(n)%2, n = n/(1+I), n = (n-1)/(1+I));); return (1); \\ Rémy Sigrist, Nov 16 2022

More terms from Charles R Greathouse IV, Nov 15 2022

A342932 The unique sequence {a(1), a(2), a(3), a(4), ...} of digits 1, 2, or 3 such that the number a(n)a(n-1)...a(2)a(1), read in base 6, is divisible by 3^n.

Original entry on oeis.org

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

Offset: 1

Eugen Ionascu, Mar 29 2021

The distribution seems to be uniform but random (empirical observation).

To prove that such a digit sequence exists and is unique is a good (but uncommon) example of a proof by induction.

			3 is divisible by 3^1;
13_6 = 1*6 + 3 = 9, which is divisible by 3^2,
213_6 = 2*6^2 + 1*6 + 3 = 81, which is divisible by 3^3.

Rémy Sigrist, Colored scatterplot of (n, 3*o(n)-n) for = 1..1000000 (where o corresponds to the ordinal transform of the sequence)
Eugen Ionascu, Mathematica File

Cf. A023396, A053312, A126933.

nd[n_] := Module[{k, i, s, ss, L, a}, L = Array[f, n]; f[1] = 3;
  Do[s = Sum[6^(k - 1)*f[k], {k, 1, i - 1}];
   ss = Mod[2^(i - 1)*s/3^(i - 1), 3];
   If[ss == 0, f[i] = 3, If[ss == 1, f[i] = 2, f[i] = 1]], {i, 2, n}];
  s = Sum[6^(k - 1)*f[k], {k, 1, n}];
  {L, s/3^n}]

{ q=0; t=1; for (n=1, 105, print1 (d=[3,1,2][1+lift(-q/Mod(t,3))]", "); q=(t*d+q)/3; t*=2) } \\ Rémy Sigrist, Apr 15 2021

n, div, divnum = 0, 1, 0
while n < 87:
    div, a = 3*div, 1
    while (a*6**n+divnum)%div != 0:
        a = a+1
    divnum, n = divnum+a*6**n, n+1
    print(a, end=', ') # A.H.M. Smeets, Apr 13 2021

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User: Eugen Ionascu

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)(1+i) + c(2)(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.

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A342932 The unique sequence {a(1), a(2), a(3), a(4), ...} of digits 1, 2, or 3 such that the number a(n)a(n-1)...a(2)a(1), read in base 6, is divisible by 3^n.

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cp's OEIS Frontend

User: Eugen Ionascu

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)*(1+i) + c(2)*(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.

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Extensions

A342932 The unique sequence {a(1), a(2), a(3), a(4), ...} of digits 1, 2, or 3 such that the number a(n)a(n-1)...a(2)a(1), read in base 6, is divisible by 3^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A358432 Nonnegative integers m which can be represented using only 0's and 1's in the complex base 1+i, i.e., m = c(0) + c(1)(1+i) + c(2)(1+i)^2 + ... where each coefficient c(k) is either 0 or 1.