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A376548 a(1)=1,a(2)=2. Let p be the smallest prime factor of j=a(n-1), then for n>2 a(n) is the smallest novel multiple of Sopfr(j) if j is in A046363, or of p if it is not.

%I A376548 #9 Oct 16 2024 21:35:19
%S A376548 1,2,4,6,5,10,7,14,8,12,21,3,9,15,18,16,20,22,13,26,24,28,11,33,27,30,
%T A376548 32,34,19,38,36,40,44,42,46,48,55,25,35,45,66,50,52,17,51,39,54,77,49,
%U A376548 56,65,60,58,31,62,64,68,70,72,74,76,23,69,57,63,78,80,91
%N A376548 a(1)=1,a(2)=2. Let p be the smallest prime factor of j=a(n-1), then for n>2 a(n) is the smallest novel multiple of Sopfr(j) if j is in A046363, or of p if it is not.
%C A376548 If j=a(n-1) is in A046363 then a(n) = A001414(j) = prime q, and a(n+1) is the least novel multiple of q. Otherwise a(n) is the least novel multiple of the smallest prime factor of j. After a prime term a(n) the sequence produces a string of terms each divisible by the smallest prime factor of a(n+1) until arriving at a term in A046363, whereupon a new prime appears and the process repeats.
%C A376548 Conjectured to be a permutation of the positive integers A000027, in which the primes do not appear in order (prime order starts:2,5,7,3,13,11,19,17,31,23,43..).
%H A376548 Michael De Vlieger, <a href="/A376548/b376548.txt">Table of n, a(n) for n = 1..10000</a>
%H A376548 Michael De Vlieger, <a href="/A376548/a376548.png">Log log scatterplot of a(n)</a>, n = 1..2^12, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not squarefree.
%H A376548 Michael De Vlieger, <a href="/A376548/a376548_1.png">Log log scatterplot of a(n)</a>, n = 1..2^20, with the same color function as immediately above. Note various trajectories of primes.
%e A376548 a(2)=2 is not a term in A046360, and has smallest prime factor 2, so a(3) = 4, the least novel multiple of 2. Likewise a(4)=6 since a(3)=4 is not in A046360 and the smallest prime factor of 4 is 2.
%e A376548 a(4)=6 is a term in A046360, so a(5)=A001414(6)=5.
%e A376548 a(6)=10 since 5 is the smallest prime factor of 5, and 10 is the smallest novel multiple of 5.
%e A376548 If a(n-1) = prime p, a(n) is the least novel multiple of p, for example a(12) = 3 and since a(4) = 6 it follows that a(13) = 9. Likewise a(19) = 13, and since no prior term is divisible by 13, a(20) = 36.
%t A376548 nn = 120; c[_] := False; m[_] := 1;
%t A376548 a[1] = 1; j = a[2] = 2; c[1] = c[2] = True; m[1] = m[2] = 2;
%t A376548 f[x_] := f[x] = Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]];
%t A376548 Do[(If[PrimeQ[#2], k = #2, k = #1]; While[c[k*m[k]], m[k]++]; k *= m[k]) & @@
%t A376548   {FactorInteger[j][[1, 1]], f[j]};
%t A376548   Set[{a[i], c[k], j}, {k, True, k}], {i, 3, nn}];
%t A376548 Array[a, nn] (* _Michael De Vlieger_, Sep 28 2024 *)
%Y A376548 Cf. A000027, A001414, A300813.
%K A376548 nonn
%O A376548 1,2
%A A376548 _David James Sycamore_, Sep 27 2024