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A330530 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by 4.

%I A330530 #17 Aug 20 2023 04:35:46
%S A330530 1,4,2,6,8,3,12,5,16,7,20,9,24,10,14,18,22,26,28,11,32,13,36,15,40,17,
%T A330530 44,19,48,21,52,23,56,25,60,27,64,29,68,30,34,38,42,46,50,54,58,62,66,
%U A330530 70,72,31,76,33,80,35,84,37,88,39,92,41,96,43,100,45,104
%N A330530 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by 4.
%C A330530 For any k > 0, let f_k be the lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by k:
%C A330530 - in particular:
%C A330530      f_1 = f_2 = A000027,
%C A330530      f_3 = A006368,
%C A330530      f_4 = a (this sequence),
%C A330530      f_6 = A330531,
%C A330530 - f_k is a permutation of the natural numbers,
%C A330530 - f_k(1) = 1, f_k(2) = max(2, k),
%C A330530 - if k is prime, then f_k corresponds to the integers that are not multiple of k interspersed with the integers that are multiple of k.
%C A330530 Apparently:
%C A330530 - for m > 0, the m-th run of consecutive terms such that gcd(a(n), 4) = 2 has A153893(m+1) terms,
%C A330530 - for m > 1, the m-th run of consecutive terms such that gcd(a(n), 4) = 1 or 4 has A068156(m+1) terms.
%H A330530 Rémy Sigrist, <a href="/A330530/b330530.txt">Table of n, a(n) for n = 1..10000</a>
%H A330530 Rémy Sigrist, <a href="/A330530/a330530.png">Colored scatterplot of the first 10000 terms</a>
%H A330530 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A330530 The first terms, alongside their product with the next term, are:
%e A330530   n   a(n)  a(n)*a(n+1)
%e A330530   --  ----  -----------
%e A330530    1     1            4
%e A330530    2     4            8
%e A330530    3     2           12
%e A330530    4     6           48
%e A330530    5     8           24
%e A330530    6     3           36
%e A330530    7    12           60
%e A330530    8     5           80
%e A330530    9    16          112
%e A330530   10     7          140
%o A330530 (PARI) s=0; v=1; for (n=1, 10 000, print (n " " v); s+=2^v; for (w=1, oo, if (!bittest(s,w) && (v*w)%4==0, v=w; break)))
%Y A330530 Cf. A006368, A068156, A153893, A330531 (f_6), A330576 (inverse).
%K A330530 nonn,easy,look
%O A330530 1,2
%A A330530 _Rémy Sigrist_, Dec 17 2019