cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A125755 Integer solutions corresponding to the terms in A125754.

Original entry on oeis.org

-2, 4, -2, -11, -14, -20, -32, 4, -2, 16, 13, -2, 40, 4, -11, -14, -20, -2, -32, 4, 16, 13, 40, 4, -2, 40, 58, 49, 76, 43, 61, 70, 58, 112, 85, 67, 148, 112, 61, 88, 106, 88, 160, 133, 196, 304, 277, 148, 130, 184, 256, 202, 184, 400, 832, 4, 16, 4, -2, 13
Offset: 1

Views

Author

David Applegate, Feb 02 2007

Keywords

Links

Crossrefs

Cf. A125754.

A125626 Numbers n whose reverse binary representation has the following property: let a 0 mean "halving" and a 1 mean "k -> 3k+1". The number describes an operation k -> f_n(k). If the equation f_n(k) = k has a positive solution, n is a term in the sequence.

Original entry on oeis.org

4, 8, 16, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 140, 144, 145, 146, 148, 152, 160, 161, 162, 164, 168, 176, 192, 193, 194, 196, 200, 208, 224, 256, 257, 258, 259, 260
Offset: 4

Views

Author

Nicholas Sanders (gummybean(AT)gmail.com), Jan 27 2007

Keywords

Comments

The terms in this sequence have the following characterization. Suppose the binary expansion of n contains i 1's and j 0's. Then it is easy to see that n is in the sequence if and only if 3^i < 2^j, or i/j < log 2 / log 3 = 0.630929753... - David Applegate and N. J. A. Sloane, Feb 01 2007
Note that f_n(x) is always a linear function of x.
The reverse binary expansions of the first few terms are:
001
0001
00001
000001
100001
010001
001001
000101
000011
0000001
1000001
0100001
0010001
0001001
0000101
0000011
00000001
10000001
01000001
11000001
00100001
...
Could be used in conjunction with the Collatz (or 3x+1) conjecture. If the positive solution k is an integer (most are not) then a cycle exists. If this cycle does not contain a 1 and the sequence of steps agrees with what Collatz's rule tells you to do when you start with k, then the Collatz conjecture would be false.

Examples

			Consider the term 200: its binary representation is 11001000. Reversing this gives 00010011. We solve (3*(3*(((3*(((k/2)/2)/2)+1)/2)/2)+1)+1) = k and find k = 40. Since k is positive, 200 is a member of the sequence.

Crossrefs

For the values of n for which the fixed point k is a positive (or any) integer, see A125754-A125757.
Cf. A112695, A125710, A125711.

Programs

  • C
    #include  #include  #include  void multiply(float *coef, float *cons) { (*coef) *= 3; (*cons) = 3*(*cons)+1; } void divide(float *coef, float *cons) { (*coef) /= 2; (*cons) /= 2; } int main() { int a, b, c, n; float coef, cons, final; char data[30], sequence[30]; for (a = 1; a < 500; a++) { coef = 1; cons = 0; c = a; sequence[0] = ''; for (b = 1; b < 12; b++) //12 is arbitrary; it allows for "a" up to 2^12 { if (c != 0) { if (c % 2) { sprintf(sequence, "%s1", sequence); multiply(&coef, &cons); } else { sprintf(sequence, "%s0", sequence); divide(&coef, &cons); } c = trunc(c/2); } else break; } if (coef >= 1.0) { coef -= 1.0; cons *= -1.0; } else coef = 1.0-coef; final = cons/coef; if (final > 0) { sprintf(data, "%10.3f %s %d ", final, sequence, a); printf(data); } } return 0; }

A125756 Numbers n whose reverse binary representation has the following property: let a 0 mean "halving" and a 1 mean "k -> 3k+1". The number describes an operation k -> f_n(k). If the equation f_n(k) = k has a positive integer solution, n is a term in the sequence.

Original entry on oeis.org

4, 36, 140, 145, 200, 292, 1059, 1136, 1553, 1800, 2340, 4150, 4274, 4297, 4308, 4389, 4433, 4490, 4634, 4696, 4705, 4741, 4804, 4876, 5133, 5164, 5218, 5254, 5400, 5409, 5668, 5712, 5761, 6244, 6290, 6312, 6448, 6466, 6662, 6800, 6976
Offset: 1

Views

Author

David Applegate, Feb 02 2007

Keywords

Comments

Suggested by A125626.
Note that f_n(x) is always a linear function of x.

Links

Crossrefs

Cf. A125626, A125755, A125754, A125757, A125710, A125711.

A125758 Numbers congruent to 4 or 7 (mod 9).

Original entry on oeis.org

4, 7, 13, 16, 22, 25, 31, 34, 40, 43, 49, 52, 58, 61, 67, 70, 76, 79, 85, 88, 94, 97, 103, 106, 112, 115, 121, 124, 130, 133, 139, 142, 148, 151, 157, 160, 166, 169, 175, 178, 184, 187, 193, 196, 202, 205, 211, 214, 220, 223, 229, 232, 238, 241, 247, 250, 256, 259, 265, 268
Offset: 1

Views

Author

N. J. A. Sloane and David Applegate, Feb 02 2007

Keywords

Comments

For a given integer m, write its binary representation in reverse order, as in A125626, A125754, etc.; let a 0 mean "halving" and a 1 mean "k -> 3k+1". Then m specifies an operation on real numbers given by k -> f_m(k). Suppose the equation f_m(k) = k has a positive integer solution for some m. Then we conjecture that the values of k are precisely the terms of this sequence.
25 is a term because we have 25 -> 76 -> 38 -> 19 -> 58 -> 29 -> 88 -> 44 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 25.
In other words, we conjecture that this sequence coincides with A125757 sorted and with duplicates removed.

Links

Crossrefs

Cf. A125626, A125754, A125755, A125756, A125757, A125710, A125711.

Programs

  • Mathematica
    Select[Range[300],MemberQ[{4,7},Mod[#,9]]&]  (* Harvey P. Dale, Mar 12 2011 *)

Formula

From R. J. Mathar, Apr 03 2009: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 9.
a(n) + a(n+1) = A017185(n).
G.f.: x*(4+3*x+2*x^2)/((1+x)*(x-1)^2). (End)
E.g.f.: 2 + ((9*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 21 2022
Showing 1-4 of 4 results.

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