A297075 - OEIS

%I #28 Feb 01 2024 10:44:48

%S 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 125,19,100,21,108,22,121,23,128,20,216,26,144,24,169,29,196,30,200,

%U 31,225,33,243,28,256,34,288,35,289,37,324,38,343,39,361,40,400

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

%C 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 Equivalently, for any n > 0, A297404(a(n)) AND A297404(a(n+1)) = 0 (where AND denotes the bitwise AND operator).

%C This sequence is a permutation of the natural numbers, with inverse A297403.

%C 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 Lexicographically earliest sequence of distinct numbers such that gcd(A181819(a(n)), A181819(a(n+1))) = 1. - _Peter Munn_, Oct 02 2023

%C From _Peter Munn_, Jan 25 2024: (Start)

%C 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 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 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 (End)

%H Rémy Sigrist, <a href="/A297075/b297075.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A297075/a297075.png">Colored logarithmic scatterplot of the first 15000 terms</a>

%H Rémy Sigrist, <a href="/A297075/a297075.txt">C++ program for A297075</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e The first terms, alongside the corresponding sets of prime exponents, are:

%e   n       a(n)    Set of prime exponents of a(n)

%e   --      ----    ------------------------------

%e    1       1      {}

%e    2       2      {1}

%e    3       4      {2}

%e    4       3      {1}

%e    5       8      {3}

%e    6       5      {1}

%e    7       9      {2}

%e    8       6      {1, 1}

%e    9      16      {4}

%e   10       7      {1}

%e   11      25      {2}

%e   12      10      {1, 1}

%e   13      27      {3}

%e   14      11      {1}

%e   15      32      {5}

%e   16      12      {2, 1}

%e   17      64      {6}

%e   18      13      {1}

%e   19      36      {2, 2}

%e   20      14      {1, 1}

%t 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 (C++) See Links section.

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

%K nonn

%O 1,2

%A _Rémy Sigrist_, Dec 25 2017