This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A375281 #15 Jul 01 2025 08:54:52
%S A375281 1,2,6,3,10,9,12,4,15,8,18,5,24,16,30,7,60,14,90,20,27,36,25,42,35,66,
%T A375281 175,84,40,81,48,32,45,64,54,70,21,110,63,120,28,105,22,210,11,420,33,
%U A375281 140,72,125,96,128,75,126,50,150,49,180,56,165,98,240,77,270
%N A375281 Lexicographically earliest infinite sequence of distinct positive integers such that for all n > 1 a(n-1), a(n) do not have the same number of distinct prime divisors, whereas the squarefree kernel of their product is a primorial number.
%C A375281 In other words omega(a(n-1)) != omega(a(n)) whilst rad(a(n-1)*a(n)) is a term in A002110, where omega is A001221 and rad is A007947. In general, primes appear consequent to primorial terms (a(11,12) = 18,5 is an exception). Conjectured to be a permutation of the positive integers, with primes in order.
%C A375281 From _Michael De Vlieger_, Aug 19 2024: (Start)
%C A375281 Let r be squarefree and define lineage S_r to be the list of numbers k such that rad(k) = r. Then S_r = r*R_r, where R_r is the list of numbers m such that rad(m) | r, hence k = m*r. R_r is the sorted, vectorized tensor product of prime divisor power ranges { p^j : p | r, j >= 0 }. The smallest term of S_r is r itself; it is the only squarefree number in the lineage S_r. S_1 = {1}, but S_r, r > 1 is infinite.
%C A375281 By construction, this sequence features k in S_r in order on account of the squarefree kernel constraint. Example, S_6 = A033845; S_6(1) = a(3) = 6, S_6(2) = a(7) = 12, S_6(3) = a(11) = 18, etc.
%C A375281 Corollary: Prime powers p^j appear in order.
%C A375281 Therefore, if this sequence is a permutation of natural numbers, we should see all squarefree numbers in this sequence, and all k in S_r should appear in order. (End)
%H A375281 Michael De Vlieger, <a href="/A375281/b375281.txt">Table of n, a(n) for n = 1..10000</a>
%H A375281 Michael De Vlieger, <a href="/A375281/a375281.png">Log log scatterplot of a(n)</a>, n = 1..10000, 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 additionally signifying powerful numbers that are not prime powers.
%H A375281 Michael De Vlieger, <a href="/A375281/a375281_1.png">Fan style binary tree showing n for a(n) = A019565(i)</a>, i = 0..511, with a color function where red = 1 and magenta = 10000.
%H A375281 Michael De Vlieger, <a href="/A375281/a375281_2.png">Log log scatterplot of a(n)</a>, n = 1..10^4, with a color function showing omega(a(n)) <= 1 in black, omega(a(n)) = 2 in red, ..., and omega(a(n)) = 9 in blue.
%e A375281 The sequence starts a(1) = 1, a(2) = 2 since this is the earliest pair of positive integers with differing numbers of distinct prime divisors (omega(1) = 0, omega(2) = 1) such that the squarefree kernel rad(1*2) = 2 = A002110(1) is a primorial number.
%e A375281 a(3) = 6 since 3,4,5, all have omega = 1, the same number of distinct prime divisors as 2, but omega(6) = 2 and rad(2*6) = 6 = A002110(2).
%t A375281 Clear[c]; nn = 64; s = {1, 2}; kk = Length[s]; u = 1;
%t A375281 c[_] := False; P = FoldList[Times, 1, Prime@ Range[60] ];
%t A375281 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A375281 MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, s];
%t A375281 Set[{i, j}, {a[kk - 1], a[kk]}]; While[c[u], u++];
%t A375281 Do[k = u;
%t A375281 While[Or[c[k], FreeQ[P, rad[j*k]], PrimeNu[k] == PrimeNu[j] ], k++];
%t A375281 Set[{a[n], c[k], i, j}, {k, True, j, k}];
%t A375281 If[k == u, While[c[u], u++] ], {n, kk + 1, nn}];
%t A375281 Array[a, nn] (* _Michael De Vlieger_, Aug 19 2024 *)
%Y A375281 Cf. A001221, A002110, A007947.
%K A375281 nonn
%O A375281 1,2
%A A375281 _David James Sycamore_, Aug 09 2024
%E A375281 More terms from _Michael De Vlieger_, Aug 19 2024