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A364154 Lexicographically earliest sequence of distinct positive integers such that a(n) is least novel multiple m of the product of all primes less than the greatest prime factor of a(n-1) which do not divide a(n-1); a(1) = 1.

%I A364154 #17 Jul 15 2023 08:48:57
%S A364154 1,2,3,4,5,6,7,30,8,9,10,12,11,210,13,2310,14,15,16,17,30030,18,19,
%T A364154 510510,20,21,40,24,22,105,26,1155,28,45,32,23,9699690,25,36,27,34,
%U A364154 15015,38,255255,42,35,48,29,223092870,31,6469693230,33,70,39,770,51,10010
%N A364154 Lexicographically earliest sequence of distinct positive integers such that a(n) is least novel multiple m of the product of all primes less than the greatest prime factor of a(n-1) which do not divide a(n-1); a(1) = 1.
%C A364154 It follows from the definition that the sequence is infinite, and that the records (outside of the first 7 terms) are all primorial numbers, meaning that it grows very quickly.
%C A364154 When there are no primes less than the greatest prime factor of a(n-1) which do not divide a(n-1) then m is the least novel multiple of 1, the empty product, and therefore a(n) = u, the least unused number in the sequence so far. The only way a prime can enter the sequence is as u. When a(n-1) = prime(k), a(n) is A002110(k-1), and any primorial term is followed by u. Thus: prime —> primorial —> u.
%C A364154 Sequence is a permutation of the positive integers since by the definition no number appears more than once and m = 1 eventually introduces any number not already placed by the first part of the definition (m > 1).
%H A364154 Michael De Vlieger, <a href="/A364154/b364154.txt">Table of n, a(n) for n = 1..4992</a>
%H A364154 Michael De Vlieger, <a href="/A364154/a364154.png">Log log scatterplot of log_10(a(n))</a>, n = 1..2^16.
%H A364154 Michael De Vlieger, <a href="/A364154/a364154_1.png">Plot p^e | a(n) at (x,y) = (n, pi(p))</a> for n = 1..2^11, with a color function representing e = 1 as black, e = 2 as red, ..., the maximum value of e in the dataset as magenta. Under the plot, we indicate the empty product in black, primes in red, composite prime powers in gold, squarefree composites in green, and all other numbers in blue.
%e A364154 a(1) = 1 and there are no primes < 1 which divide 1 therefore m = 1 so a(2) = 2, the least unused number. Likewise a(3) = 3.
%e A364154 a(4) = 2*2 = 4 since 2 is the only prime < 3 which does not divide 3 and 2 has already occurred.
%e A364154 Since a(7) = 7, a(8) = 2*3*5 = 30.
%t A364154 nn = 120; c[_] := False; m[_] := 1; a[1] = j = 1; c[1] = True;
%t A364154 Do[k = Times @@ Complement[Prime@ Range[PrimePi@ Last[#] - 1], #] &[
%t A364154    FactorInteger[j][[All, 1]] ];
%t A364154  While[c[k m[k]], m[k]++]; k *= m[k];
%t A364154  Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}];
%t A364154 Array[a, nn]
%o A364154 (PARI) lista(nn) = my(c, m, v=List([1, 2])); for(k=3, nn, c=m=1; forprime(p=2, vecmax(factor(v[k-1])[, 1]), if(v[k-1]%p, m*=p)); while(setsearch(Set(v), c*m), c++); listput(v, c*m)); Vec(v) \\ _Jinyuan Wang_, Jul 11 2023
%Y A364154 Cf. A002110, A083720, A351495, A359804, A363195.
%K A364154 nonn
%O A364154 1,2
%A A364154 _David James Sycamore_, Jul 11 2023
%E A364154 More terms from _Jinyuan Wang_, Jul 11 2023