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A330919 Lexicographically earliest sequence of distinct squarefree numbers such that for any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) is a prime number.

%I A330919 #16 May 03 2020 13:50:28
%S A330919 1,2,6,3,15,5,10,30,210,42,14,7,21,105,35,70,770,110,22,11,33,66,330,
%T A330919 165,55,385,77,154,462,231,1155,2310,30030,2730,390,78,26,13,39,195,
%U A330919 65,130,910,182,91,273,546,6006,858,286,143,429,2145,715,1430,4290
%N A330919 Lexicographically earliest sequence of distinct squarefree numbers such that for any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) is a prime number.
%C A330919 In other words, consecutive terms differ exactly by one prime factor.
%C A330919 This sequence has strong connections with A163252:
%C A330919 - here consecutive terms differ by one prime factor, there by one binary digit,
%C A330919 - for any n > 0, A163252(n-1) encodes in binary form the prime numbers appearing in a(n).
%C A330919 Odd indexed terms have an even number of prime factors and vice versa.
%C A330919 For any prime number p: as there are only finitely many squarefree numbers with greatest prime factor < p, eventually the sequence contains a multiple of p.
%H A330919 Rémy Sigrist, <a href="/A330919/b330919.txt">Table of n, a(n) for n = 1..10000</a>
%H A330919 Rémy Sigrist, <a href="/A330919/a330919.gp.txt">PARI program for A330919</a>
%F A330919 a(n) = A019565(A163252(n-1)).
%F A330919 A087207(a(n)) = A163252(n-1).
%e A330919 The first terms, alongside their prime factors, are:
%e A330919   n   a(n)  prime factors
%e A330919   --  ----  -------------
%e A330919    1     1
%e A330919    2     2  2
%e A330919    3     6  2, 3
%e A330919    4     3     3
%e A330919    5    15     3, 5
%e A330919    6     5        5
%e A330919    7    10  2,    5
%e A330919    8    30  2, 3, 5
%e A330919    9   210  2, 3, 5, 7
%e A330919   10    42  2, 3,    7
%o A330919 (PARI) See Links section.
%Y A330919 Cf. A019565, A087207, A163252.
%K A330919 nonn,look
%O A330919 1,2
%A A330919 _Rémy Sigrist_, May 02 2020