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A375024 a(n) is the length of the largest sequence S of distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2, and two consecutive terms in S have some common prime factor.

%I A375024 #6 Jul 28 2024 12:52:38
%S A375024 1,1,1,2,2,3,3,3,3,4,4,4,4,4,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,
%T A375024 10,10,10,13,13,16,16,16,16,19,19,19,19,19,19
%N A375024 a(n) is the length of the largest sequence S of distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2, and two consecutive terms in S have some common prime factor.
%C A375024 Sequences like A280864 can be split into segments of consecutive terms with properties similar to the sequences S that we are considering here.
%H A375024 Rémy Sigrist, <a href="/A375024/a375024.c++.txt">C++ program</a>
%F A375024 a(n) <= A373797(n).
%F A375024 a(p) = a(p-1) for any prime number p.
%e A375024 Some solutions for small n:
%e A375024   n   a(n)  Solution S
%e A375024   --  ----  --------------------------------------------------------------
%e A375024    1     1  1
%e A375024    4     2  2,4
%e A375024    6     3  2,6,3
%e A375024   10     4  3,6,10,5
%e A375024   15     7  3,6,10,15,12,14,7
%e A375024   21    10  3,6,10,15,12,14,21,18,20,5
%e A375024   33    13  3,6,10,15,12,14,21,18,22,33,24,20,5
%e A375024   35    16  3,6,10,15,12,14,21,18,20,35,28,22,33,24,26,13
%e A375024   39    19  3,6,10,15,12,14,21,18,20,35,28,22,33,24,26,39,36,34,17
%e A375024   45    22  5,10,6,15,20,12,21,14,18,33,22,24,39,26,36,45,40,28,35,30,42,7
%o A375024 (C++) // See Links section.
%Y A375024 Cf. A280864, A283832, A373797.
%K A375024 nonn,more
%O A375024 1,4
%A A375024 _Rémy Sigrist_, Jul 28 2024