A370822 Lexicographically earliest sequence of positive integers such that all equal terms appear at mutually coprime indices.

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 4, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 7, 14, 1, 15, 1, 16, 8, 17, 3, 18, 1, 19, 10, 20, 1, 21, 1, 22, 11, 23, 1, 24, 2, 25, 13, 26, 1, 27, 6, 28, 14, 29, 1, 30, 1, 31, 16, 32, 7, 33, 1, 34, 17, 35, 1, 36, 1, 37, 19

Offset: 1

Author

Neal Gersh Tolunsky, Mar 02 2024

Keywords

• nonn

Comments

See A279119 for the same sequence with numbers including 0.

See A055396 for a similar sequence where all equal terms share a factor > 1.

Examples

			a(4)=2 because if we had a(4)=1, then i=2 and i=4, which are not coprime indices, would have the same value 1. So a(4)=2, which is a first occurrence.
a(9)=2 because if we had a(9)=1, i=3 and i=9, would have the same value despite not being coprime indices. a(9) can be 2 because the only other index with a 2 is a(4)=2 and 4 is coprime to 9.
a(15)=4 because 4 is the smallest value such that every previous index at which a 4 occurs is coprime to i=15. In this case, 4 has only occurred at i=8 and 8 is coprime to 15.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A100480, A279119, A370408.

Programs

Python

from math import gcd, lcm
from itertools import combinations as C, count, islice
def agen(): # generator of terms
    yield from [1, 1, 1]
    lcms = {1: 6}
    for n in count(4):
        an = next(an for an in count(1) if an not in lcms or gcd(lcms[an], n) == 1)
        yield an
        if an not in lcms: lcms[an] = n
        else: lcms[an] = lcm(lcms[an], n)
print(list(islice(agen(), 75))) # Michael S. Branicky, Mar 02 2024

Formula

a(n) = 1 + A279119(n). - Rémy Sigrist, Mar 04 2024

Extensions

a(22) and beyond from Michael S. Branicky, Mar 02 2024