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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.

%I A297075 #32 Mar 24 2025 22:34:50
%S A297075 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,
%T A297075 125,19,100,21,108,22,121,23,128,20,216,26,144,24,169,29,196,30,200,
%U A297075 31,225,33,243,28,256,34,288,35,289,37,324,38,343,39,361,40,400
%N A297075 Lexicographically earliest sequence of distinct positive numbers such that the prime factorizations of two consecutive terms never share a prime exponent >= 1.
%C A297075 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).
%C A297075 Equivalently, for any n > 0, A297404(a(n)) AND A297404(a(n+1)) = 0 (where AND denotes the bitwise AND operator).
%C A297075 This sequence is a permutation of the natural numbers, with inverse A297403.
%C A297075 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).
%C A297075 Lexicographically earliest sequence of distinct numbers such that gcd(A181819(a(n)), A181819(a(n+1))) = 1. - _Peter Munn_, Oct 02 2023
%C A297075 From _Peter Munn_, Jan 25 2024: (Start)
%C A297075 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.
%C A297075 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).
%C A297075 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.
%C A297075 (End)
%H A297075 Rémy Sigrist, <a href="/A297075/b297075.txt">Table of n, a(n) for n = 1..10000</a>
%H A297075 Rémy Sigrist, <a href="/A297075/a297075.png">Colored logarithmic scatterplot of the first 15000 terms</a>
%H A297075 Rémy Sigrist, <a href="/A297075/a297075.txt">C++ program for A297075</a>
%H A297075 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A297075 The first terms, alongside the corresponding sets of prime exponents, are:
%e A297075   n       a(n)    Set of prime exponents of a(n)
%e A297075   --      ----    ------------------------------
%e A297075    1       1      {}
%e A297075    2       2      {1}
%e A297075    3       4      {2}
%e A297075    4       3      {1}
%e A297075    5       8      {3}
%e A297075    6       5      {1}
%e A297075    7       9      {2}
%e A297075    8       6      {1, 1}
%e A297075    9      16      {4}
%e A297075   10       7      {1}
%e A297075   11      25      {2}
%e A297075   12      10      {1, 1}
%e A297075   13      27      {3}
%e A297075   14      11      {1}
%e A297075   15      32      {5}
%e A297075   16      12      {2, 1}
%e A297075   17      64      {6}
%e A297075   18      13      {1}
%e A297075   19      36      {2, 2}
%e A297075   20      14      {1, 1}
%t A297075 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 *)
%o A297075 (C++) // See Links section.
%Y A297075 Cf. A001694 (numbers in odd bisection), A036967, A038109, A052485 (numbers in even bisection), A181819, A297403 (inverse), A297404.
%K A297075 nonn
%O A297075 1,2
%A A297075 _Rémy Sigrist_, Dec 25 2017