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.

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A358668 a(n) is the least m such that A359194^k(m) = n for some k >= 0 (where A359194^k denotes the k-th iterate of A359194).

Original entry on oeis.org

0, 0, 2, 3, 4, 5, 3, 7, 8, 9, 7, 11, 12, 3, 14, 11, 11, 17, 11, 19, 20, 14, 12, 23, 3, 12, 26, 12, 28, 29, 11, 12, 32, 33, 12, 35, 36, 11, 38, 12, 29, 41, 42, 28, 44, 45, 12, 47, 48, 26, 50, 51, 12, 53, 54, 3, 56, 26, 23, 59, 60, 12, 62, 26, 26, 65, 26, 67, 68
Offset: 0

Views

Author

Rémy Sigrist, Dec 22 2022

Keywords

Comments

See A359214 for the corresponding minimal k's.

Examples

			The orbit of 0 under repeated application of A359194 is:
    0, 1, 0, ...
So a(0) = a(1) = 0.
The orbit of 2 under repeated application of A359194 is:
    2, 1, 0, 1, 0, ...
So a(2) = 2.
The orbit of 3 under repeated application of A359194 is:
    3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0, 1, 0, ...
So a(3) = a(6) = a(13) = a(24) = a(55) = a(90) = a(241) = a(300) = a(123) = a(142) = a(85) = 3.

Links

Crossrefs

Cf. A070167, A359194, A359214.

Programs

  • PARI
    See Links section.

Formula

a(n) <= n.

A359214 a(n) is the least k >= 0 such that A359194^k(A358668(n)) = n (where A359194^k denotes the k-th iterate of A359194).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 4, 3, 0, 5, 0, 0, 1, 74, 0, 3, 7, 0, 1, 0, 0, 1, 5, 0, 0, 6, 0, 0, 2, 0, 77, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 4, 0, 9, 1, 0, 0, 75, 0, 7, 6, 0, 8, 0, 0, 1, 0, 0, 76, 0, 0, 1, 5418, 0, 1, 0, 0, 2, 0, 0
Offset: 0

Views

Author

Rémy Sigrist, Dec 22 2022

Keywords

Examples

			The orbit of 0 under repeated application of A359194 is:
    0, 1, 0, ...
So a(0) = 0, a(1) = 1.
The orbit of 2 under repeated application of A359194 is:
    2, 1, 0, 1, 0, ...
So a(2) = 0.
The orbit of 3 under repeated application of A359194 is:
    3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0, 1, 0, ...
So a(3) = 0, a(6) = 1, a(13) = 2, a(24) = 3, a(55) = 4, etc.

Links

Crossrefs

Cf. A358668, A359194, A359226.
Cf. A343858 (smallest numbers inside cyclic trajectories of the generalized Collatz function bx+c).

Programs

  • Mathematica
    nn = 83; c[] = -1; c[0] = 0; f[n] := FromDigits[BitXor[1, IntegerDigits[3*n, 2]], 2]; Do[(MapIndexed[If[c[#1] == -1, Set[c[#1], First[#2] - 1]] &, #]; -1 + Length[#]) &@ NestWhileList[f, n, c[#] == -1 && # > 1 &], {n, 0, nn}]; Array[c, nn] (* Michael De Vlieger, Dec 23 2022 *)
  • PARI
    See Links section.

Formula

a(n) = 0 iff A358668(n) = n.
a(3*n+2) = 0. - Thomas Scheuerle, Dec 22 2022

A359259 a(n) is the least k such that A359194(k) = A032766(n).

Original entry on oeis.org

1, 0, 4, 9, 3, 8, 18, 7, 17, 6, 16, 37, 15, 36, 14, 35, 13, 34, 12, 33, 11, 32, 74, 31, 73, 30, 72, 29, 71, 28, 70, 27, 69, 26, 68, 25, 67, 24, 66, 23, 65, 22, 64, 149, 63, 148, 62, 147, 61, 146, 60, 145, 59, 144, 58, 143, 57, 142, 56, 141, 55, 140, 54, 139
Offset: 0

Views

Author

Rémy Sigrist, Dec 23 2022

Keywords

Comments

The binary expansion of numbers m such that A359194(m) = A032766(n):
- starts with zero or more occurrences of "10",
- followed by a "0" when the binary expansion of a(n) starts with zero or more occurrences of "10" followed by "11",
- ends with the binary expansion of a(n) (assuming that 0 has an empty binary expansion).

Examples

			The first terms, alongside the binary expansions of A032766(n) and a(n), are:
  n   a(n)  bin(A032766(n))  bin(a(n))
  --  ----  ---------------  ---------
   0     1                0          1
   1     0                1          0
   2     4               11        100
   3     9              100       1001
   4     3              110         11
   5     8              111       1000
   6    18             1001      10010
   7     7             1010        111
   8    17             1100      10001
   9     6             1101        110
  10    16             1111      10000
  11    37            10000     100101

Crossrefs

Cf. A032766, A359194.

Programs

  • PARI
    a(n) = { if (n<=1, return (1-n), n+=n\2; for (x=2+exponent(n), oo, my (k=bitneg(n,x)); if (k%3==0, return (k/3)))) }

A359266 Numbers k such that A359194(k) > k.

Original entry on oeis.org

0, 3, 6, 7, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109
Offset: 1

Views

Author

Rémy Sigrist, Dec 23 2022

Keywords

Comments

The first run of consecutive values has length 1 and initial value 0.
For k > 1, the k-th run of consecutive values has length A000975(k-1) and initial value A005578(k+1).

Examples

			A359194(43) = 126 > 43, so 43 belongs to this sequence.

Crossrefs

Cf. A000975, A005578, A359194, A359267 (complement).

Programs

  • PARI
    is(n) = if(n, bitneg(3*n, exponent(3*n)+1), 1) > n

A359267 Numbers k such that A359194(k) < k.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 19, 20, 21, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143
Offset: 1

Views

Author

Rémy Sigrist, Dec 23 2022

Keywords

Comments

The first run of consecutive values has length 2 and initial value 1.
For k > 1, the k-th run of consecutive values has length A005578(k) and initial value 2^k.

Examples

			A359194(42) = 1 < 42, so 42 belongs to this sequence.

Crossrefs

Cf. A005578, A359194, A359266 (complement).

Programs

  • PARI
    is(n) = if(n, bitneg(3*n, exponent(3*n)+1), 1) < n

A359268 a(n) is the least k such that A359194(k) = A359194(n).

Original entry on oeis.org

0, 1, 0, 3, 4, 1, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 3, 4, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 6, 7, 8, 9, 0, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69
Offset: 0

Views

Author

Rémy Sigrist, Dec 23 2022

Keywords

Examples

			The first terms, alongside A359194(n), are:
  n   a(n)  A359194(n)
  --  ----  ----------
   0     0           1
   1     1           0
   2     0           1
   3     3           6
   4     4           3
   5     1           0
   6     6          13
   7     7          10
   8     8           7
   9     9           4
  10     0           1
  11    11          30
  12    12          27

Links

Crossrefs

Cf. A002450, A020988, A359194, A359259.

Programs

  • PARI
    b(n) = if(n, bitneg(3*n, exponent(3*n)+1), 1)
a(n) = { my (v=b(n)); while (n, my (x=exponent(n)); if (b(n-2^x)==v, n-=2^x, break)); return (n) }

Formula

a(n) = 0 iff n belongs to A020988.
a(n) = 1 iff n belongs to A002450 \ {0}.
Previous Showing 11-16 of 16 results.

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