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%I #10 Jun 20 2024 22:50:36
%S 1,2,3,6,4,5,12,7,10,9,8,15,14,11,30,21,16,20,18,25,24,27,32,35,33,42,
%T 40,22,84,45,28,49,60,36,48,64,50,54,70,44,39,105,66,13,140,99,26,210,
%U 55,52,126,110,56,63,90,72,75,81,80,77,78,65,154,51,130,231
%N a(1,2) = 1,2; i = a(n-2), j = a(n-1). For n > 2 a(n) is least novel k (with A001221(k) != w if A001221(i) = A001221(j) = w), such that A007947(i*j*k) is a term in A002110.
%C In other words, a(n) is the smallest novel k satisfying the above constraint on omega(k) such that the squarefree kernel of i*j*k is a primorial number. Conjectured to be a permutation of the positive integers (A000027), with primes in order.
%H Michael De Vlieger, <a href="/A373323/b373323.txt">Table of n, a(n) for n = 1..5002</a>
%e a(1,2) = 1,2 and a(3) = 3 is least k such that rad(1*2*3) = 6 is primorial, with omega(1,2,3) = 0,1,1 satisfied.
%e a(2,3) = 2,3 with omega(2,3) = 1,1 so a(4) cannot be 4 (even though rad(2*3*4) = 6), because omega(4) = 1. Therefore a(4) = 6, the least k satisfying the omega condition (omega(6) = 2) such that rad(2*3*6) = 6 is primorial.
%e a(3,4) = 3,6 with omega(3,6) = 1,2 therefore a(5) = 4 since rad(3*6*4) = 6 and no smaller term is available at this point (omega constraint is not invoked).
%t nn = 120; c[_] := False;
%t f[x_] := Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
%t Array[Set[{a[#], c[#]}, {#, True}] &, 3];
%t Set[{i, j, u, v, w}, {2, 3, 4, 1, 1}];
%t Do[m = i*j; k = u;
%t While[Or[c[k], Equal @@ {v, w, Set[x, PrimeNu[k]]}, ! f[m*k]], k++];
%t Set[{a[n], c[k], i, j, v, w}, {k, True, j, k, w, x}];
%t If[k == u, While[c[u], u++]], {n, 4, nn}];
%t Array[a, nn] (* _Michael De Vlieger_, Jun 03 2024 *)
%Y Cf. A000027, A001221, A002110, A007947, A373326.
%K nonn,new
%O 1,2
%A _David James Sycamore_, Jun 01 2024
%E More terms from _Michael De Vlieger_, Jun 03 2024
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