A297403 -id:A297403 - cp's OEIS frontend

A297075 Lexicographically earliest sequence of distinct positive numbers such that the prime factorizations of two consecutive terms never share a prime exponent >= 1.

1, 2, 4, 3, 8, 5, 9, 6, 16, 7, 25, 10, 27, 11, 32, 12, 64, 13, 36, 14, 49, 15, 72, 17, 81, 18, 125, 19, 100, 21, 108, 22, 121, 23, 128, 20, 216, 26, 144, 24, 169, 29, 196, 30, 200, 31, 225, 33, 243, 28, 256, 34, 288, 35, 289, 37, 324, 38, 343, 39, 361, 40, 400

Offset: 1

Rémy Sigrist, Dec 25 2017

nonn

For any n > 0, if a prime number p divides a(n) and a prime number q divides a(n+1), then the p-adic valuation of a(n) differs from the q-adic valuation of a(n+1).
Equivalently, for any n > 0, A297404(a(n)) AND A297404(a(n+1)) = 0 (where AND denotes the bitwise AND operator).
This sequence is a permutation of the natural numbers, with inverse A297403.
The curves visible in the logarithmic scatterplot of the first terms seems to be related to a(n) belonging to A038109 and to A052485 (see Links section).
Lexicographically earliest sequence of distinct numbers such that gcd(A181819(a(n)), A181819(a(n+1))) = 1. - Peter Munn, Oct 02 2023
From Peter Munn, Jan 25 2024: (Start)
The sequence bisections might be characterized as being monotonic with interruptions. The major interruptions are apparent from the coloring in the author's 15000 term logarithmic scatterplot -- they occur where the occurrence of terms belonging to A038109 switches between the bisections.
Other interruptions are too small to be seen in the scatterplot. Some relate to numbers that have both the square of a prime and cube of a prime as a unitary divisor (a subset of A038109).
Two such terms are a(4154) = 1350 and a(4156) = 1368, interrupting the even bisection's monotonicity after a(4152) = 1380. These 3 terms are each followed by a 4-full number (A036967): a(4153) = 1185921, a(4155) = 1229312, a(4157) = 1250000. Then we see an odd bisection interruption with a(4159) = 1191016.
(End)

			The first terms, alongside the corresponding sets of prime exponents, are:
  n       a(n)    Set of prime exponents of a(n)
  --      ----    ------------------------------
   1       1      {}
   2       2      {1}
   3       4      {2}
   4       3      {1}
   5       8      {3}
   6       5      {1}
   7       9      {2}
   8       6      {1, 1}
   9      16      {4}
  10       7      {1}
  11      25      {2}
  12      10      {1, 1}
  13      27      {3}
  14      11      {1}
  15      32      {5}
  16      12      {2, 1}
  17      64      {6}
  18      13      {1}
  19      36      {2, 2}
  20      14      {1, 1}

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 15000 terms
Rémy Sigrist, C++ program for A297075
Index entries for sequences that are permutations of the natural numbers

Cf. A001694 (numbers in odd bisection), A036967, A038109, A052485 (numbers in even bisection), A181819, A297403 (inverse), A297404.

Nest[Append[#, Block[{k = 3, m = FactorInteger[#[[-1]] ][[All, -1]]}, While[Nand[FreeQ[#, k], ! IntersectingQ[m, FactorInteger[k][[All, -1]]]], k++]; k]] &, {1, 2}, 61] (* Michael De Vlieger, Dec 29 2017 *)

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A297075 Lexicographically earliest sequence of distinct positive numbers such that the prime factorizations of two consecutive terms never share a prime exponent >= 1.

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