A380728 For n a power of 2, a(n) = n. Otherwise a(n) is the smallest number not yet in the sequence which is coprime to n and has the same binary weight as n.

1, 2, 5, 4, 3, 17, 11, 8, 10, 9, 7, 65, 14, 13, 23, 16, 6, 257, 21, 33, 19, 25, 15, 1025, 22, 35, 29, 37, 27, 43, 47, 32, 20, 129, 26, 4097, 28, 41, 46, 513, 38, 67, 30, 49, 53, 39, 31, 16385, 44, 69, 58, 73, 45, 71, 59, 81, 77, 51, 55, 83, 62, 61, 95, 64, 12

Offset

1,2

Comments

Self inverse sequence with fixed points on powers of 2 (similar to A005940). Records subsequence (after 1,2) set by odd numbers with binary weight = 2 (see A000051, for n >= 2, and also A048578).

Conjectured to be a permutation of the natural numbers (primes not in order).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Log log scatterplot of log_2(a(n)), n = 2..2^14.

Formula

a(a(n)) = n for all n.

Example

a(1) = 1, the smallest novel number coprime to 1 and having same binary weight (1)
a(3) = 5, since weight(3) = weight(5) = 2 and gcd(2,5) = 1, and 5 is least such number.
a(5) = 3 (sequence is self inverse).

Mathematica

nn = 2^13; c[_] := False; u = 1;
f[x_] := f[x] = DigitCount[x, 2, 1];
Reap[Do[w = f[n];
  Which[w == 1, k = n,
    And[w == 2, EvenQ[n]],
      k = 3; While[Or[c[k], ! CoprimeQ[k, n]], k = 2*(k - 1) + 1],
    True, k = u; While[Or[c[k], ! CoprimeQ[k, n], w != f[k]], k++] ];
  Sow[k]; c[k] = True;
If[k == u, While[c[u], u++]], {n, nn}] ][[-1, 1]] (* Michael De Vlieger, Feb 02 2025 *)

Crossrefs

Cf. A000051, A005940, A048578, A085731.

Sequence in context: A266416 A266401 A083798 * A197377 A211247 A021397

Adjacent sequences: A380725 A380726 A380727 * A380729 A380730 A380731

Keyword

nonn,base

Author

David James Sycamore, Jan 31 2025

Extensions

More terms from Michael De Vlieger, Feb 02 2025.

Status

approved