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.

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, 175, 84, 40, 81, 48, 32, 45, 64, 54, 70, 21, 110, 63, 120, 28, 105, 22, 210, 11, 420, 33, 140, 72, 125, 96, 128, 75, 126, 50, 150, 49, 180, 56, 165, 98, 240, 77, 270

Offset: 1

Author

David James Sycamore, Aug 09 2024

Keywords

• nonn

Comments

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.

From Michael De Vlieger, Aug 19 2024: (Start)

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.

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.

Corollary: Prime powers p^j appear in order.

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)

Examples

			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.
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).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), 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.
• Michael De Vlieger, Fan style binary tree showing n for a(n) = A019565(i), i = 0..511, with a color function where red = 1 and magenta = 10000.
• Michael De Vlieger, Log log scatterplot of a(n), 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.

Crossrefs

Cf. A001221, A002110, A007947.

Programs

Mathematica

Clear[c]; nn = 64; s = {1, 2}; kk = Length[s]; u = 1;
c[_] := False; P = FoldList[Times, 1, Prime@ Range[60] ];
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, s];
Set[{i, j}, {a[kk - 1], a[kk]}]; While[c[u], u++];
Do[k = u;
  While[Or[c[k], FreeQ[P, rad[j*k]], PrimeNu[k] == PrimeNu[j] ], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, k}];
  If[k == u, While[c[u], u++] ], {n, kk + 1, nn}];
Array[a, nn] (* Michael De Vlieger, Aug 19 2024 *)

Extensions

More terms from Michael De Vlieger, Aug 19 2024