A369317 -id:A369317 - cp's OEIS frontend

A369277 Distinct values of A369317, in order of appearance.

1, 3, 7, 5, 15, 9, 31, 21, 11, 13, 63, 17, 51, 127, 85, 33, 73, 255, 27, 45, 511, 65, 341, 23, 107, 29, 19, 189, 195, 25, 1023, 273, 69, 81, 455, 129, 585, 79, 93, 819, 207, 121, 243, 2047, 1365, 279, 635, 443, 889, 465, 4095, 257, 1419, 1677, 1057, 313, 1335

Offset: 1

Rémy Sigrist, Jan 20 2024

All terms are even.
This sequence is infinite as it contains A126646.
Will every odd number appear in the sequence?
Empirically, each odd number, say v, appears in A369317, and the first index is of the form v*2^k - 1 for some k > 0 (see Example section).

			The first terms, alongside their index m in A369317, in decimal and in binary, are:
  n   a(n)  m     bin(a(n))  bin(m)
  --  ----  ----  ---------  ------------
   1     1     1          1             1
   2     3     5         11           101
   3     7    27        111         11011
   4     5    39        101        100111
   5    15   119       1111       1110111
   6     9   287       1001     100011111
   7    31   495      11111     111101111
   8    21   671      10101    1010011111
   9    11   703       1011    1010111111
  10    13   831       1101    1100111111
  11    63  2015     111111   11111011111
  12    17  2175      10001  100001111111

Rémy Sigrist, PARI program
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A126646, A369317.

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A369281 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, A091255(a(n), a(n+1)) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 8, 9, 13, 10, 19, 12, 21, 15, 14, 17, 16, 23, 22, 25, 18, 31, 20, 35, 24, 37, 26, 27, 32, 29, 28, 33, 38, 39, 41, 30, 47, 34, 49, 40, 55, 36, 59, 42, 43, 44, 45, 50, 51, 52, 53, 56, 57, 61, 46, 67, 48, 69, 54, 73, 58, 79, 60, 81, 62

Offset: 1

Rémy Sigrist, Jan 18 2024

In other words, the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive terms are coprime.
As the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive integers are not necessarily coprime (see A369317-A369318), the present sequence does not equal the identity map.
This sequence is a permutation of the positive integers with inverse A369282:
- we can always extend the sequence with some term of A014580 not yet in the sequence, hence the sequence is infinite, and all terms of A014580 appear in the sequence, in ascending order,
- for any k > 0, the first term >= A014580(k) is precisely A014580(k),
- if a(n) = A014580(k) for some n and the least value not among the first n terms, say u, is less than A014580(k), then a(n+1) = u,
- and eventually every integer will appear in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 2500 terms.
Rémy Sigrist, PARI program.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

See A369293 for a similar sequence.

Cf. A014580, A091255, A369282 (inverse), A369317-A369318.

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A369318 Numbers k such that A091255(k, k + 1) <> 1.

Original entry on oeis.org

5, 9, 17, 23, 27, 29, 33, 35, 39, 45, 53, 57, 65, 71, 77, 83, 85, 89, 95, 101, 105, 107, 113, 119, 125, 129, 135, 139, 141, 149, 153, 159, 165, 169, 177, 179, 183, 189, 195, 197, 201, 209, 215, 221, 223, 225, 231, 237, 245, 249, 251, 257, 259, 263, 269, 277

Offset: 1

Rémy Sigrist, Jan 19 2024

Equivalently, numbers k such that A369317(k) <> 1.

Two consecutive integers are always coprime, however the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive integers are not necessarily coprime, hence this sequence.

			The first terms, alongside the correspond GF(2)[X]-polynomials, are:
  n   a(n)  P(a(n))              P(a(n)+1)            gcd(P(a(n)), P(a(n)+1))
  --  ----  -------------------  -------------------  -----------------------
   1     5  X^2 + 1              X^2 + X              X + 1
   2     9  X^3 + 1              X^3 + X              X + 1
   3    17  X^4 + 1              X^4 + X              X + 1
   4    23  X^4 + X^2 + X + 1    X^4 + X^3            X + 1
   5    27  X^4 + X^3 + X + 1    X^4 + X^3 + X^2      X^2 + X + 1
   6    29  X^4 + X^3 + X^2 + 1  X^4 + X^3 + X^2 + X  X + 1
   7    33  X^5 + 1              X^5 + X              X + 1
   8    35  X^5 + X + 1          X^5 + X^2            X^2 + X + 1
   9    39  X^5 + X^2 + X + 1    X^5 + X^3            X^2 + 1
  10    45  X^5 + X^3 + X^2 + 1  X^5 + X^3 + X^2 + X  X + 1

Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A091255, A369317.

is(n) = fromdigits(lift(Vec(gcd(Mod(1, 2) * Pol(binary(n)), Mod(1, 2) * Pol(binary(n+1))))), 2) != 1

cp's OEIS Frontend

A369277 Distinct values of A369317, in order of appearance.

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A369281 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, A091255(a(n), a(n+1)) = 1.

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PARI

A369318 Numbers k such that A091255(k, k + 1) <> 1.

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PARI