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A378233 a(1) = 1, a(2) = 2, let i = a(n-2), j = a(n-1). For n > 2, If i*j is a term in A055932, a(n) is the least novel multiple of the smallest prime which does not divide i*j. Otherwise a(n) is the least novel multiple of the greatest prime < Gpf(i*j) which does not divide i*j.

%I A378233 #19 Nov 24 2024 09:37:27
%S A378233 1,2,3,5,4,6,10,7,9,15,8,14,20,12,21,25,16,18,30,28,11,35,24,22,42,40,
%T A378233 33,49,45,26,44,56,50,27,63,55,32,70,36,66,77,60,13,88,84,65,99,91,75,
%U A378233 110,98,39,121,105,34,52,132,112,80,48,119,78,143,126,85,104
%N A378233 a(1) = 1, a(2) = 2, let i = a(n-2), j = a(n-1). For n > 2, If i*j is a term in A055932, a(n) is the least novel multiple of the smallest prime which does not divide i*j. Otherwise a(n) is the least novel multiple of the greatest prime < Gpf(i*j) which does not divide i*j.
%C A378233 In other words, if the product i*j of adjacent terms has primorial kernel, rad(i*j) =  A002110(k) for some k>= 0, the next term is the smallest novel multiple of prime(k+1). Otherwise a(n) is the smallest novel multiple of p, the greatest prime less than the greatest prime factor of i*j, which does not divide i*j. The definition is similar to that of A359804, the difference being in the second condition, where here the greatest non divisor prime is used instead of the smallest. Same first 9 terms as A359804, departure thereafter.
%C A378233 Fixed points occur at indices 1,2,3,6,9,18, after which it seems there are no more... Primes p = 2,3,5,7,11 occur prior to 2*p, whereas 13 occurs after 26, and 17 occurs after 34.
%C A378233 Conjectures: Sequence is a permutation of the natural numbers A000027, with primes in order. Prime powers and numbers that share the same squarefree kernel also appear in order.
%C A378233 Alternate definition: a(n) = least m*A377774(a(n-2)*a(n-1)), m >= 1, such that a(k) != a(n), k < n. - _Michael De Vlieger_, Nov 23 2024
%H A378233 Michael De Vlieger, <a href="/A378233/b378233.txt">Table of n, a(n) for n = 1..10000</a>
%H A378233 Michael De Vlieger, <a href="/A378233/a378233.png">Log log scatterplot of a(n)</a>, n = 1..1048576.
%H A378233 Michael De Vlieger, <a href="/A378233/a378233_1.png">Log log scatterplot of a(n)</a>, n = 1..16384, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple also shows numbers that are powerful but not prime powers.
%e A378233 Since rad(1*2) = 2 = A002110(1), a(3) = prime(2) = 3.
%e A378233 Then rad(2*3) = 6 = A002110(2), so a(4) = prime(3) = 5.
%e A378233 Rad(3*5) = 15, not a term in A055932, 2 is the greatest prime < 5 = Gpf(15) which does not divide 15, so a(5) = 4, the least novel multiple of 2.
%e A378233 Rad(5*4) = 10 so a(6) is 6, the least novel multiple of 3, the greatest prime < 5 which does not divide 10.
%e A378233 a(8) = 7, a(9) = 9, 7*9 = 63 is not in A055932 so a(10) = 15, the least novel multiple of 5, the greatest prime < 7 = Gpf(63) which does not divide 63. a(10) = 15 is the first departure from A359804.
%t A378233 c[_] := False; m[_] := 1; nn = 120;
%t A378233 Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j}, Range[2]];
%t A378233 Do[s = FactorInteger[i*j][[All, 1]]; h = Times @@ s;
%t A378233  If[Or[IntegerQ@ Log2[h],
%t A378233    And[EvenQ[h], Union@ Differences@ PrimePi[s] == {1}] ],
%t A378233   k = NextPrime[s[[-1]]]; While[c[k*m[k]], m[k]++]; k *= m[k],
%t A378233   k = s[[-1]]; While[Divisible[h, k = NextPrime[k, -1] ] ];
%t A378233   While[c[k*m[k]], m[k]++]; k *= m[k] ];
%t A378233  Set[{a[n], c[k], i, j}, {k, True, j, k}],
%t A378233  {n, 3, nn}];
%t A378233 Array[a, nn]
%Y A378233 Cf. A000027, A000040, A002110, A006530, A055932, A098550, A359804, A377774.
%K A378233 nonn,look
%O A378233 1,2
%A A378233 _David James Sycamore_ and _Michael De Vlieger_, Nov 20 2024