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A373323
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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.
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1
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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, 40, 22, 84, 45, 28, 49, 60, 36, 48, 64, 50, 54, 70, 44, 39, 105, 66, 13, 140, 99, 26, 210, 55, 52, 126, 110, 56, 63, 90, 72, 75, 81, 80, 77, 78, 65, 154, 51, 130, 231
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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.
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).
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MATHEMATICA
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nn = 120; c[_] := False;
f[x_] := Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
Array[Set[{a[#], c[#]}, {#, True}] &, 3];
Set[{i, j, u, v, w}, {2, 3, 4, 1, 1}];
Do[m = i*j; k = u;
While[Or[c[k], Equal @@ {v, w, Set[x, PrimeNu[k]]}, ! f[m*k]], k++];
Set[{a[n], c[k], i, j, v, w}, {k, True, j, k, w, x}];
If[k == u, While[c[u], u++]], {n, 4, nn}];
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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