A373323 - OEIS

A373323

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.

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

1,2

COMMENTS

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.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..5002

EXAMPLE

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).

MATHEMATICA

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}];

Array[a, nn] (* Michael De Vlieger, Jun 03 2024 *)

CROSSREFS

Cf. A000027, A001221, A002110, A007947, A373326.