cp's OEIS Frontend

A285296 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.

%I A285296 #21 Jun 14 2017 02:46:49
%S A285296 1,4,2,6,3,8,5,9,7,12,10,14,16,11,18,13,20,15,21,24,17,25,19,27,22,26,
%T A285296 28,23,32,29,36,30,33,39,40,31,44,34,38,42,35,45,37,48,41,49,43,50,46,
%U A285296 52,47,54,51,56,53,60,55,63,57,64,58,62,66,68,59,72,61
%N A285296 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.
%C A285296 The sequence can always be extended with a number that is not squarefree (say a multiple of 4); after a term that is not squarefree, we can extend the sequence with the least unused number; as there are infinitely many multiples of 4, this sequence is a permutation of the natural numbers (with inverse A285297).
%C A285296 Conjecturally, a(n) ~ n.
%C A285296 This sequence has similarities with A075380: here we consider the product of consecutive terms, there the sum of consecutive terms.
%C A285296 For any k>0, let b_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^k for some prime p; in particular we have:
%C A285296 - b_1 = A000027 (the natural numbers),
%C A285296 - b_2 = a (this sequence),
%C A285296 - b_3 = A285299,
%C A285296 - b_4 = A285386,
%C A285296 - b_5 = A285417.
%C A285296 For any k>0, b_k is a permutation of the natural numbers.
%C A285296 For any k>0, b_k(1)=1 and b_k(2)=2^k.
%C A285296 Graphically, the sequences from b_2 to b_5 differ.
%H A285296 Rémy Sigrist, <a href="/A285296/b285296.txt">Table of n, a(n) for n = 1..2000</a>
%H A285296 Rémy Sigrist, <a href="/A285296/a285296.gp.txt">PARI program for A285296</a>
%H A285296 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A285296 The first terms, alongside the primes p such that p^2 divides a(n)*a(n+1), are:
%e A285296 n       a(n)    p
%e A285296 --      ----    -
%e A285296 1       1       2
%e A285296 2       4       2
%e A285296 3       2       2
%e A285296 4       6       3
%e A285296 5       3       2
%e A285296 6       8       2
%e A285296 7       5       3
%e A285296 8       9       3
%e A285296 9       7       2
%e A285296 10      12      2
%e A285296 11      10      2
%e A285296 12      14      2
%e A285296 13      16      2
%e A285296 14      11      3
%e A285296 15      18      3
%e A285296 16      13      2
%e A285296 17      20      2, 5
%e A285296 18      15      3
%e A285296 19      21      2, 3
%e A285296 20      24      2
%Y A285296 Cf. A000027, A075380, A285297 (inverse).
%K A285296 nonn
%O A285296 1,2
%A A285296 _Rémy Sigrist_, Apr 16 2017