A065190 -id:A065190 - cp's OEIS frontend

A355166 Lexicographically earliest sequence of distinct positive integers such for any n > 0, n and a(n) are coprime and have no common 1-bits in their binary expansions.

2, 1, 4, 3, 8, 17, 16, 5, 20, 21, 32, 19, 18, 33, 64, 7, 6, 13, 12, 9, 10, 41, 40, 35, 34, 37, 68, 65, 66, 97, 96, 11, 14, 25, 24, 67, 26, 73, 80, 23, 22, 85, 84, 81, 82, 129, 128, 71, 72, 69, 76, 75, 74, 137, 136, 131, 70, 133, 132, 193, 130, 257, 256, 15, 28

Offset: 1

Rémy Sigrist, Jun 22 2022

This sequence combines features of A065190 and of A238757.
This sequence is a self-inverse permutation of the nonnegative integers, without fixed points.
This sequence is well defined:
- if n is odd, then we can extend the sequence with a power of 2 > n,
- if n < 2^k is even, then we can extend the sequence with a prime number of the form 1 + t*2^k (Dirichlet's theorem on arithmetic progressions guarantees us that there is an infinity of such prime numbers).
When n is odd, a(n) is even and vice-versa.

			The first terms, alongside binary expansions and distinct prime factors, are:
  n   a(n)  bin(n)  bin(a(n))  dpf(n)  dpf(a(n))
  --  ----  ------  ---------  ------  ---------
   1     2       1         10  {}      {2}
   2     1      10          1  {2}     {}
   3     4      11        100  {3}     {2}
   4     3     100         11  {2}     {3}
   5     8     101       1000  {5}     {2}
   6    17     110      10001  {2, 3}  {17}
   7    16     111      10000  {7}     {2}
   8     5    1000        101  {2}     {5}
   9    20    1001      10100  {3}     {2, 5}
  10    21    1010      10101  {2, 5}  {3, 7}

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A065190, A238757, A352633.

PARI
```
See Links section.
```

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    aset, mink = set(), 1
    for n in count(1):
        an = mink
        while an in aset or n&an or gcd(n, an)!=1: an += 1
        aset.add(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Jun 22 2022

A380671 a(n) is the smallest number not yet in the sequence which is coprime to n and shares at least one decimal digit with n.

Original entry on oeis.org

1, 21, 13, 41, 51, 61, 17, 81, 19, 11, 10, 23, 3, 15, 14, 31, 7, 71, 9, 27, 2, 25, 12, 29, 22, 63, 20, 83, 24, 37, 16, 33, 32, 35, 34, 43, 30, 39, 38, 47, 4, 121, 36, 45, 44, 49, 40, 85, 46, 53, 5, 55, 50, 59, 52, 57, 56, 65, 54, 67, 6, 69, 26, 141, 58, 161, 60

Offset: 1

David James Sycamore and Michael De Vlieger, Jan 30 2025

Like A065190 but with the extra condition that n and a(n) must have at least one decimal digit in common. Definition implies that a(1) = 1 is the only fixed point. Let a(n) be even then 2|a(n) -> 2!|n-> 2|(n+1)->2!|a(n+1), therefore there are no consecutive even terms. Let [n] = |n - a(n)|, then it follows from the coprime conditions of the definition that n, a(n), [n] are pairwise coprime. Sequence is conjectured to be a permutation of the natural numbers with [n] < 100 (= base^2).

			a(1) = 1 since 1 is the smallest novel number prime to 1 and sharing a digit with it so a(2) = 21 because digit 2 is shared, Gcd(2,21) = 1 and there is no smaller number with this property. a(7) = 17 implies a(17) = 7 (self inverse property).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A065190.

nn = 120; c[_] := True; u = 1;
Reap[Do[s = Union@ IntegerDigits[n]; k = u;
  While[
    Nand[c[k], IntersectingQ[s, IntegerDigits[k]], CoprimeQ[n, k]],
    k++];
  Sow[k]; c[k] = False;
  If[k == u, While[! c[u], u++]], {n, nn}] ][[-1, 1]]

a(a(n)) = n for all n (sequence is self inverse).

cp's OEIS Frontend

A355166 Lexicographically earliest sequence of distinct positive integers such for any n > 0, n and a(n) are coprime and have no common 1-bits in their binary expansions.

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