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A387080 a(1)=1, a(2)=3; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

%I A387080 #11 Aug 19 2025 20:15:25
%S A387080 1,3,6,4,2,8,10,5,15,12,9,18,16,14,7,21,24,22,20,25,30,28,26,13,39,36,
%T A387080 34,32,38,19,57,54,52,50,48,46,44,42,40,35,45,33,27,51,17,68,66,64,62,
%U A387080 60,58,56,49,63,69,23,92,90,88,86,84,82,80,78,76,74,72,70
%N A387080 a(1)=1, a(2)=3; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.
%C A387080 This is a variant of A386482 that begins with 1,3 instead of 1,2.
%H A387080 Michael De Vlieger, <a href="/A387080/b387080.txt">Table of n, a(n) for n = 1..10000</a>
%H A387080 Michael De Vlieger, <a href="/A387080/a387080.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A387080 Michael De Vlieger, <a href="/A387080/a387080_2.png">Log log scatterplot of a(n) in red and A386482(n) in blue</a>, n = 1..2^20.
%H A387080 Michael De Vlieger, <a href="/A387080/a387080_1.png">Log log scatterplot of a(n)</a>, n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where the latter represents powerful numbers that are not prime powers.
%t A387080 Block[{c, j, k, m, p, r, nn},
%t A387080   nn = 2^12; c[_] := False; m[_] := 1; j = 2; c[1] = c[2] = True; r = 1;
%t A387080   {1}~Join~Monitor[Most@ Reap[Do[
%t A387080     If[PrimePowerQ[j],
%t A387080       Set[{p, k, m}, {#1, #1^(#2 - 1), #1^(#2 - 1)}] & @@
%t A387080         FactorInteger[j][[1]]; While[And[c[k*p], k != 0], k--];
%t A387080         If[k == 0, k = m; While[c[k*p], k++]]; k *= p,
%t A387080       k = j - 1; While[And[Or[c[k], CoprimeQ[j, k]], k != 1], k--];
%t A387080         If[k == 1, k += j; While[Or[c[k], CoprimeQ[j, k] ], k++] ] ];
%t A387080     If[Mod[j, 2] == Mod[k, 2], r++, Sow[r]; r = 1];
%t A387080     Set[{c[k], j}, {True, k}], {n, 3, nn}] ][[-1, 1]], n] ]
%Y A387080 Cf. A386482.
%K A387080 nonn
%O A387080 1,2
%A A387080 _Geoffrey Caveney_ and _Michael De Vlieger_, Aug 16 2025