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A369825 a(1,2) = 1,2; thereafter let i = a(n-2) and r(n) = a(1)*a(2)*...*a(n-1): a(n) is the smallest novel multiple m of w = rad(r(n))/rad(i), where rad is A007947.

%I A369825 #23 Mar 02 2024 13:30:14
%S A369825 1,2,4,3,6,8,5,15,12,10,20,9,18,30,25,7,42,60,35,14,24,45,70,28,21,75,
%T A369825 40,56,63,90,50,49,84,120,55,77,126,150,110,154,105,135,22,308,210,
%U A369825 165,11,98,420,330,33,91,910,660,66,182,455,495,132,364,1365,825
%N A369825 a(1,2) = 1,2; thereafter let i = a(n-2) and r(n) = a(1)*a(2)*...*a(n-1): a(n) is the smallest novel multiple m of w = rad(r(n))/rad(i), where rad is A007947.
%C A369825 Rad(r(n)) is always a primorial (A002110), and there are two distinct ways a prime p > 2 can enter the sequence:
%C A369825 (i). Directly: rad(i) = rad(r(n)) implies a(n) is the least unused term, conjectured to be the smallest prime p not already in the sequence. In this case no prior term is divisible by p. This happens for 3, 5, 7. It is not known if this ever happens again (thought to be unlikely).
%C A369825 (ii). Indirectly: rad(i) and rad(r(n)) are consecutive primorials whose quotient is prime p = gpd(rad(r(n)). This implies that p has already entered the sequence as divisor of a previous (composite) term, since then p|rad(r(n)), and r(n) is the product of all prior terms, so p must be a factor of at least one of them.
%C A369825 In the first case m is the least novel multiple of 1, and rad(r(n)) increments to the next primorial at the point of entry a(n) = p. In the second case rad(r(n)) increments (to the next primorial; gpf = p) at the earliest term a(t); t < n where p | a(t). Example: a(35) = 55, the earliest term divisible by 11. We see six more composite terms divisible by 11, before finally a(47) = 11.
%C A369825 Prime terms occur in order as shown in Example, but only as far as 61. Primes do not appear in order for n > 667722.
%C A369825 For n < 667722, with r(A) = P(k+1) following prime(k+1) | a(A) for A < n, we admit a(n-2) = P(k) and a(n) = prime(k+1) where n < B such that r(B) = P(k+2) following prime(k+2) | a(B). For sufficiently small n, primes a(n) = prime(k+1) follow primorials a(n-2) = P(k) in the sequence.
%C A369825 Primorials and primes decouple such that a(676472) = P(18), but a(676474) = 4757 = prime(19)*prime(20). This is the result of r(n) increasing twice (at n = 253724 and 667722), offering a greater degree of freedom for a(676474) than for terms that follow previous instances of primorials in the sequence. We also have a(9061722) = 73, following a(9061720) = P(23)/73. Primorials a(n) = P(k) do not appear in order for n > 681764.
%C A369825 Powerful numbers in the sequence are sparse, since they require w to appear m times such that m*w is powerful. Only 10 powerful numbers appear for n <= 2^24. Even after 2^24 terms, 36 is still missing. Therefore the sequence is predominantly of weak numbers.
%C A369825 It is not known if this sequence could be a permutation of A000027.
%H A369825 Michael De Vlieger, <a href="/A369825/b369825.txt">Table of n, a(n) for n = 1..10000</a>
%H A369825 Michael De Vlieger, <a href="/A369825/a369825.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A369825 Michael De Vlieger, <a href="/A369825/a369825_1.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, accentuating primorials in large green, and powerful numbers that are not prime powers in purple.
%H A369825 Michael De Vlieger, <a href="/A369825/a369825_2.png">Plot p^k | a(n) at (x,y) = (n, pi(p))</a> for n = 1..2^11, 12X vertical exaggeration, with a color function showing k = 1 in black, k = 2 in red, ... maximum value of k in reference range in magenta. The color bar under the plot indicates numbers as immediately above, red = prime, etc.
%H A369825 Michael De Vlieger, <a href="/A369825/a369825_3.png">Plot a(n) at (x,y) = (x mod 1024, -floor(x/1024))</a> for n = 1..2^20, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, the latter color indicating powerful numbers that are not prime powers. Indicates large-scale pattern of prime power decomposition.
%H A369825 Michael De Vlieger, <a href="/A369825/a369825.txt">Remarks on this sequence, data regarding primes and primorials, and cellular automaton like behavior</a>
%e A369825 a(3) = 4 because rad(1*2)/rad(1) = 2 and 4 is the least novel multiple of 2.
%e A369825 a(4) = 3 because rad(1*2*4)/rad(2) = 1, and 3 is the least novel multiple of 1.
%e A369825 a(5) = 6 since rad(1*2*4*3)/rad(4) = 3 and 6 is the least novel multiple of 3.
%e A369825 a(6) = 8, the least novel multiple of 2, since rad(1*2*4*3*6)/rad(3) = 2.
%e A369825 a(7) = 5 since rad(1*2*4*3*6*8)/rad(6) = 1 and 5 is the least novel multiple of 1.
%e A369825 a(8) = 15 since rad(m)/rad(i) = 30/2 = 15, which has not appeared previously.
%e A369825 Table of n, a(n) for primes (in 3rd column, i means rad(i) = rad(a(r)); ii means rad(i) properly divides rad(r(n)), and both rad(i) and rad(r(n)) are (for the data shown here) consecutive primorials:
%e A369825        2  2 (given)
%e A369825        4  3   i
%e A369825        7  5   i
%e A369825       16  7   i
%e A369825       47 11  ii
%e A369825       96 13  ii
%e A369825      193 17  ii
%e A369825      476 19  ii
%e A369825      697 23  ii
%e A369825     1168 29  ii
%e A369825     1349 31  ii
%e A369825     4613 37  ii
%e A369825     8898 41  ii
%e A369825    19728 43  ii
%e A369825    40553 47  ii
%e A369825    49054 53  ii
%e A369825    63802 59  ii
%e A369825   240925 61  ii
%e A369825   681766 71  ii
%e A369825  2191325 79  ii
%e A369825  9061722 73  ii
%e A369825 13178788 89  ii
%e A369825 26120340 97  ii
%t A369825 nn = 1000; c[_] := False; m[_] := 1;
%t A369825 f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
%t A369825 Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; i = 1; j = p = 2;
%t A369825 Do[(While[c[Set[k, #  m[#]]], m[#]++]) &[p/i];
%t A369825   Set[{a[n], c[k], i, j, p}, {k, True, f[j], k, f[p*k]}], {n, 3, nn}];
%t A369825 Array[a, nn] (* _Michael De Vlieger_, Feb 03 2024 *)
%Y A369825 Cf. A000027, A002110, A007947.
%K A369825 nonn
%O A369825 1,2
%A A369825 _David James Sycamore_ and _Michael De Vlieger_ Feb 02 2024