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

%I A370822 #20 Mar 06 2024 04:44:30
%S A370822 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,
%T A370822 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,
%U A370822 29,1,30,1,31,16,32,7,33,1,34,17,35,1,36,1,37,19
%N A370822 Lexicographically earliest sequence of positive integers such that all equal terms appear at mutually coprime indices.
%C A370822 See A279119 for the same sequence with numbers including 0.
%C A370822 See A055396 for a similar sequence where all equal terms share a factor > 1.
%H A370822 Michael S. Branicky, <a href="/A370822/b370822.txt">Table of n, a(n) for n = 1..10000</a>
%F A370822 a(n) = 1 + A279119(n). - _Rémy Sigrist_, Mar 04 2024
%e A370822 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.
%e A370822 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.
%e A370822 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.
%o A370822 (Python)
%o A370822 from math import gcd, lcm
%o A370822 from itertools import combinations as C, count, islice
%o A370822 def agen(): # generator of terms
%o A370822     yield from [1, 1, 1]
%o A370822     lcms = {1: 6}
%o A370822     for n in count(4):
%o A370822         an = next(an for an in count(1) if an not in lcms or gcd(lcms[an], n) == 1)
%o A370822         yield an
%o A370822         if an not in lcms: lcms[an] = n
%o A370822         else: lcms[an] = lcm(lcms[an], n)
%o A370822 print(list(islice(agen(), 75))) # _Michael S. Branicky_, Mar 02 2024
%Y A370822 Cf. A100480, A279119, A370408.
%K A370822 nonn
%O A370822 1,4
%A A370822 _Neal Gersh Tolunsky_, Mar 02 2024
%E A370822 a(22) and beyond from _Michael S. Branicky_, Mar 02 2024