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A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).

%I A361593 #14 Apr 19 2023 19:00:47
%S A361593 1,2,3,6,11,10,9,30,7,46,83,34,163,70,267,20,357,322,699,1378,2399,
%T A361593 2238,6015,5326,13579,6230,25135,106,31471,14178,45755,15234,75167,
%U A361593 68078,8341,151586,228005,193966,573557,248882,1016405,306474,1571761,361830,2240065,1043414,3645309,3464394
%N A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).
%C A361593 This is a variation of A359557 where the previous three terms are added instead of two. Unlike A359557 the terms here do no rapidly reach a regime where all terms share one or more prime factors, and it is unknown if this ever occurs.
%H A361593 Michael De Vlieger, <a href="/A361593/b361593.txt">Table of n, a(n) for n = 1..750</a>
%H A361593 Michael De Vlieger, <a href="/A361593/a361593.png">Scatterplot of Log_10(a(n))</a>, n = 1..750, showing records in red.
%H A361593 Michael De Vlieger, <a href="/A361593/a361593_1.png">Log log scatterplot of log_10(a(n))</a>, n = 1..600, showing primes in red, prime powers in gold, and squarefree composites in green.
%e A361593 a(6) = 10 as a(3) + a(4) + a(5) = 3 + 6 + 11 = 20 = 2*2*5, and the smallest unused number containing 2 and 5 as factors is 10.
%t A361593 nn = 120; c[_] = False; q[_] = 1;
%t A361593 f[n_] := Times @@ FactorInteger[n][[All, 1]]; t = 3;
%t A361593 Array[Set[{a[#], c[#]}, {#, True}] &, t]; Set[{i, j, k, x}, {a[t - 2],
%t A361593    a[t - 1], a[t], f[a[t - 2] + a[t - 1] + a[t]]}];
%t A361593 Do[m = q[x];
%t A361593   While[c[x m], m++];
%t A361593   m *= x; While[c[x q[x]], q[x]++];
%t A361593   Set[{a[n], c[m], i, j, k, x}, {m, True, j, k, m, f[j + k + m]}], {n,
%t A361593 t + 1, nn}]; Array[a, nn] (* _Michael De Vlieger_, Mar 20 2023 *)
%Y A361593 Cf. A359557, A359256, A027748, A064413, A352867, A007947.
%K A361593 nonn
%O A361593 1,2
%A A361593 _Scott R. Shannon_ and _Michael De Vlieger_, Mar 16 2023