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A374575 Lexicographically earliest infinite sequence of distinct positive integers having the property that for any triple i,j,k of consecutive terms gcd(j,k) = 1 whereas A007947(i*j*k) is a term in A002110.

%I A374575 #18 Jul 01 2025 02:20:03
%S A374575 1,2,3,4,5,6,7,10,9,8,15,14,11,30,49,12,25,16,21,20,27,28,45,22,35,18,
%T A374575 55,42,13,110,63,26,165,56,33,40,77,24,65,154,39,50,231,32,75,64,81,
%U A374575 70,99,52,105,44,91,60,121,84,125,36,175,48,143,140,51,286
%N A374575 Lexicographically earliest infinite sequence of distinct positive integers having the property that for any triple i,j,k of consecutive terms gcd(j,k) = 1 whereas A007947(i*j*k) is a term in A002110.
%C A374575 In other words a(n) is least k such that (j,k) = 1 and rad(i*j*k) is a primorial number (alternatively i*j*k is a term in A055932).
%C A374575 Conjectured to be a permutation of the positive integers with primes in order.
%H A374575 Rémy Sigrist, <a href="/A374575/b374575.txt">Table of n, a(n) for n = 1..10000</a>
%H A374575 Michael De Vlieger, <a href="/A374575/a374575.png">Log log scatterplot of a(n)</a>, n = 1..10^4, showing primes in red, proper prime powers in yellow, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, with magenta signifying powerful numbers that are not prime powers.
%H A374575 Rémy Sigrist, <a href="/A374575/a374575.gp.txt">PARI program</a>
%e A374575 Sequence starts with a(1,2,3) = {1,2,3} because this is the smallest consecutive triple of terms satisfying the definition ((2,3) = 1 and rad(1*2*3) = 6 = A002110(2)).
%e A374575 a(4) = 4 because (3,4) = 1, rad(2*3*4) = 6 = A002110(2), and 4 is the smallest unused number with this property.
%o A374575 (PARI) \\ See Links section.
%Y A374575 Cf. A002110, A007947, A055932, A374351.
%K A374575 nonn
%O A374575 1,2
%A A374575 _David James Sycamore_, Jul 11 2024
%E A374575 More terms from _Rémy Sigrist_, Aug 19 2024