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A362855 a(n) = n for n <= 3; for n > 3, a(n) is the least novel multiple of k, the product of all distinct prime factors of a(n-2) that do not divide a(n-1).

%I A362855 #22 Jun 21 2025 00:41:33
%S A362855 1,2,3,4,6,5,12,10,9,20,15,8,30,7,60,14,45,28,75,42,25,84,35,18,70,21,
%T A362855 40,63,50,105,16,210,11,420,22,315,44,525,66,140,33,280,99,350,132,
%U A362855 175,198,245,264,385,24,770,27,1540,36,1155,26,2310,13,4620,39,3080,78,1925,156,2695,234,3465,52,5775
%N A362855 a(n) = n for n <= 3; for n > 3, a(n) is the least novel multiple of k, the product of all distinct prime factors of a(n-2) that do not divide a(n-1).
%C A362855 Motivated by A362631, but instead of using one prime divisor of a(n-2) which does not divide a(n-1), the product of all such primes is used to compute a(n). - _David James Sycamore_, May 07 2023
%C A362855 From _Michael De Vlieger_, May 27 2023: (Start)
%C A362855 Primes p(k) enter the sequence in order and fairly regularly through a(20543) = p(15) = 47, immediately following primorial A002110(k-1). However, a(87723) = p(17) = 59 is the next prime to appear, following a(87722) = A002110(16).
%C A362855 Conjecture: all primes appear eventually, but not in order. (End)
%C A362855 Similar to A280866, except that the denominator here is rad(a(n-1)) instead of rad(a(n-2)). Also related to A369825. - _David James Sycamore_, Jan 27 2024
%C A362855 From _Michael De Vlieger_, Apr 23 2024: (Start)
%C A362855 Conjecture: permutation of natural numbers.
%C A362855 Conjecture: the smallest missing number is always either prime or a powerful number.
%C A362855 Primes do not appear in order; a(87723) = 59 but a(91307) = 53.
%C A362855 Powerful numbers appear in clusters, e.g., for n roughly between 91200 and 91320.
%C A362855 Though it appears primorials are always followed by primes, it is logically possible but rare that primorials can be followed by a composite number. (End)
%H A362855 Michael De Vlieger, <a href="/A362855/b362855.txt">Table of n, a(n) for n = 1..10000</a>
%H A362855 Michael De Vlieger, <a href="/A362855/a362855.png">Log log scatterplot of a(n)</a>, n = 1..701, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Powerful numbers are labeled in gold (if a prime power) or blue. Primorials P(i) = A002110(i) are labeled in green.
%H A362855 Michael De Vlieger, <a href="/A362855/a362855_2.png">Log log scatterplot of a(n)</a> for n = 1..2^20.
%H A362855 Michael De Vlieger, <a href="/A362855/a362855_3.png">Plot of k = pi(p) | a(n) at (x, y) = (n, k)</a>, n = 1..4581, with a color function representing multiplicity where black indicates 1, red = 2, etc. The bar of color at the bottom indicates primes in red, composite prime powers in gold, composite squarefree in green, and other numbers in blue.
%H A362855 Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.17445.72169">Divisibility Based Lexically Earliest Sequence with Cellular Automaton Behavior</a>, ResearchGate, 2024.
%F A362855 A007947(a(n) * a(n+1)) | A007947(a(n+1) * a(n+2)). - _Peter Munn_, Apr 18 2024
%e A362855 From _Michael De Vlieger_, Apr 23 2024: (Start)
%e A362855 Let rad(x) = A007947(x) and let P(x) = A002110(x).
%e A362855 Let S = { prime p : p | a(n-2) } and let T = { prime p : p | a(n-1) }. Then k = Product_{p in S\T} p = rad(a(n-2)*a(n-1))/rad(a(n-1)).
%e A362855 a(3) = 3 since rad(1*2)/rad(2) = 1; a(1) = 1, a(2) = 2, therefore a(3) = 3*1.
%e A362855 a(4) = 4 since rad(2*3)/rad(3) = 2; a(2) = 2, thus a(4) = 2*2.
%e A362855 a(5) = 6 since rad(3*4)/rad(4) = 6/2 = 3; a(3) = 3, thus a(5) = 2*3.
%e A362855 a(91305) = 108 and a(91306) = P(17), therefore k = 1 since rad(108) | P(17). The smallest missing number is 53, therefore a(91307) = 53*1. Related sequence A368133 = b is such that it is coincident with this sequence until b(91307) = 61, since prime(18) = 61 is the smallest prime that does not divide b(91306) = P(17). (End)
%t A362855 nn = 100; c[_] := False; m[_] := 1;
%t A362855 f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
%t A362855 Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; i = 1; j = r = 2;
%t A362855 Do[(While[c[Set[k, # m[#]]], m[#]++]) &[r/f[j]];
%t A362855   Set[{a[n], c[k], i, j, r}, {k, True, j, k, f[j*k]}], {n, 3, nn}];
%t A362855 Array[a, nn] (* _Michael De Vlieger_, Feb 21 2024 *)
%Y A362855 Cf. A002110, A007947, A280866, A306237, A362631, A368133, A369825.
%K A362855 nonn
%O A362855 1,2
%A A362855 _Michael De Vlieger_ and _David James Sycamore_, May 06 2023