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A372699 a(1) = 1, j = a(n-1). For n > 1, a(n) is the least novel m such that rad(m*j) is the smallest primorial number divisible by A007947(j).

%I A372699 #51 May 25 2024 15:42:42
%S A372699 1,2,3,4,6,5,12,10,9,8,18,15,16,24,20,27,32,36,25,30,7,60,14,45,40,48,
%T A372699 50,54,75,64,72,80,81,96,90,21,70,42,35,84,100,108,120,28,105,56,135,
%U A372699 128,144,125,150,49,180,63,140,126,160,162,200,192,225,216,240
%N A372699 a(1) = 1, j = a(n-1). For n > 1, a(n) is the least novel m such that rad(m*j) is the smallest primorial number divisible by A007947(j).
%C A372699 a(n) depends upon if rad(j) = A002110(k) for some k (equivalently A083720(j) = 1), or not. If so a(n) is least novel m such that rad(m*j) = A002110(k+1). Otherwise a(n) = least novel m such that rad(m*j) = A002110(A000720(q)), where q = gpf(j).
%C A372699 Put otherwise, if p = nextprime(q), and A = A083720, then for n > 1 if A(j) = 1, a(n) is the least novel p-smooth number divisible by p, and if A(j) = w > 1, a(n) is the least novel q-smooth number divisible by w.
%C A372699 If j is a term in A002110, a(n) = smallest prime which has not yet appeared in the sequence (e.g., 1-->2, 2-->3, 6-->5, 30-->7, 210-->11, and so on).
%C A372699 Primes are in order and if p is prime and p|a(n) there is an i <= n such that a(i) = p (no multiple of p appears prior to p). Sequence is conjectured to have "Property S" of A368900. Also, for integers x, y with x < y and rad(x) = rad(y), x appears in the sequence before y. Conjecture: Sequence is a permutation of the positive integers which preserves the above mentioned properties of A000027.
%H A372699 Michael De Vlieger, <a href="/A372699/b372699.txt">Table of n, a(n) for n = 1..10000</a>
%H A372699 Michael De Vlieger, <a href="/A372699/a372699.png">Log log scatterplot of a(n)</a>, n = 1..10800.
%H A372699 Michael De Vlieger, <a href="/A372699/a372699_1.png">Log log scatterplot of a(n)</a>, n = 1..10800, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple indicating powerful numbers that are not prime powers. Primorials P(k) = A002110(k) and primes are labeled.
%H A372699 Michael De Vlieger, <a href="/A372699/a372699_2.png">Log log scatterplot of a(n)</a>, n = 1..2^18, with a color function indicating greatest prime factor p = gpf(a(n)), with red indicating p = 2, orange p = 3, ..., magenta p = 29, and pink p = 31.
%F A372699 a(n) = A002110(k) --> a(n+1) = prime(k+1); a(n+2) = 2*a(n), a(n+3) = 2*prime(k+1) (n >= 1).
%e A372699 a(1) = 1 = A002110(0) so a(2) = 2 since rad(2*1) = 2 = A002110(1).
%e A372699 a(2) = 2 = A002110(1) so a(3) = 3 since then rad(3*2) = 6 = A002110(2).
%e A372699 a(6) = 5 is not primorial so a(7) = 12, since rad(12*5) = 30 = A002110(3) and no smaller novel number has this property.
%t A372699 nn = 120;
%t A372699 c[_] := False; m[_] := 1;
%t A372699 Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = a[2];
%t A372699 f[x_] := f[x] = FactorInteger[x][[All, 1]];
%t A372699 Do[(If[Divisible[j, Set[q, Times @@ Prime@ Range[#2]]],
%t A372699        q = Times @@ Prime@ Range[#2 + 1]];
%t A372699        k = q/#3; mm = m[k];
%t A372699        While[Or[c[k*mm], ! Divisible[q, Times @@ f[mm]]], mm++];
%t A372699        k *= mm; While[c[k*m[k]], m[k]++]) & @@
%t A372699      {#, PrimePi@ Last[#], Times @@ #} &[f[j]];
%t A372699   Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
%t A372699 Array[a, nn]
%Y A372699 Cf. A000027, A002110, A007947, A083720, A368900.
%K A372699 nonn
%O A372699 1,2
%A A372699 _David James Sycamore_ and _Michael De Vlieger_, May 14 2024
%E A372699 More terms from _Michael De Vlieger_, May 18 2024