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A379044 a(1) = 1, a(2) = 2. For n > 2 a(n) is the smallest positive integer k, not already a term in the sequence, such that A053669(k) = A006530(a(n-1)).

%I A379044 #16 Jul 21 2025 13:13:58
%S A379044 1,2,3,4,5,6,8,7,30,12,10,18,14,60,24,16,9,20,36,22,210,90,42,120,48,
%T A379044 26,2310,420,150,54,28,180,66,630,240,72,32,11,840,270,78,4620,1050,
%U A379044 300,84,330,1260,360,96,34,30030,6930,1470,390,9240,1680,450,102,60060,11550,1890,480,108,38,510510
%N A379044 a(1) = 1, a(2) = 2. For n > 2 a(n) is the smallest positive integer k, not already a term in the sequence, such that A053669(k) = A006530(a(n-1)).
%C A379044 For n > 2, a(n) is the smallest novel number whose least non divisor prime is the greatest prime divisor of a(n-1). Let p be the smallest prime which does not divide a(n-1). If rad(a(n-1)) = A002110(k), p = prime(k+1), otherwise p is the smallest prime < Gpf(a(n-1) which does not divide a(n-1).
%C A379044 The terms of the sequence can be extracted from the table in A125703, where the k-th row gives numbers whose least non divisor prime p is the k-th prime. From a(5) onwards the first appearance of q = Gpf(a(n-1)) = prime(k) gives a(n) = A002110(k-1), the first term in the k-th row of A125703. Subsequent appearances of same q selects a(n) from the same row. The primes q = Gpf(a(n-1)) decrement from term to term if rad(a(n-1) is in A002110, and occasionally we see a full cycle of q from some higher prime terminating at q = 2 (see Example). However this is rare, because of interruptions when rad(a(n-1)) is not primorial, causing q to increment before decrementing again. This retards the arrival of odd numbers in the sequence. Whereas we may suppose that all numbers in A125703 will eventually appear, the blocking of full decremetation cycles of q produces a "firewall" effect opposing this possibility. Conjecture: Full cycles of decrementing q conclude with a(n) a power of 2 (producing the next odd term).
%H A379044 Michael De Vlieger, <a href="/A379044/b379044.txt">Table of n, a(n) for n = 1..10000</a>
%H A379044 Michael De Vlieger, <a href="/A379044/a379044.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%H A379044 Michael De Vlieger, <a href="/A379044/a379044_1.png">Log log scatterplot of log_10(a(n))</a>, n = 1..2^16
%H A379044 Michael De Vlieger, <a href="/A379044/a379044_2.png">Log log scatterplot of a(n)</a>, n = 1..1024 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 purple also represents powerful numbers that are not prime powers.
%e A379044 a(3) = 3 since the greatest prime factor of a(2)is 2, and 3 is the smallest novel number whose smallest non divisor prime is 2.
%e A379044 a(5) = 5 implies a(6) = 6 because 6 is the smallest novel term whose smallest non divisor prime is 5.
%e A379044 a(16) = 16 (Gpf = 2) so a(17) = 9, the smallest novel odd (composite) number.
%e A379044 a(37) = 32 (Gpf = 2) so a(38) = 11 the smallest novel odd (prime) number.
%e A379044 a(13,14,15,16) = 14,60,24,16 with decrementing Gpf's 7,5,3,2 respectively, leading to a(17) = 9. Likewise a(33,34,35,36,37) = 66,630,240,72,32 with decrementing Gpf's 11,7,5,3,2 respectively, leading to a(38) = 11 (next odd number).
%t A379044 nn = 120; c[_] := False; m[_] := 1;
%t A379044 a[1] = 1; a[2] = j = p = 2; facs = {2}; c[1] = c[2] = True;
%t A379044 f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q];
%t A379044 Do[p = Max[facs];
%t A379044   If[IntegerQ@ Log2[j], k = m[1]; While[Or[c[k], f[k] != p], k++];
%t A379044     If[k > m[1], While[c[m[1]], m[1]++]],
%t A379044     k = Product[Prime[i], {i, -1 + PrimePi[p]}];
%t A379044       While[Or[c[#], f[#] != p] &[k*m[k]], m[k]++]; k *= m[k]];
%t A379044   Set[{a[n], c[k], j, facs},
%t A379044     {k, True, k, FactorInteger[k][[All, 1]] } ], {n, 3, nn}];
%t A379044 Array[a, nn] (* _Michael De Vlieger_, Dec 17 2024 *)
%Y A379044 Cf. A002110, A006530, A007947, A053669, A125703.
%K A379044 nonn
%O A379044 1,2
%A A379044 _David James Sycamore_, Dec 14 2024