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%I #23 Mar 06 2021 08:19:34
%S 1,2,1,3,6,2,4,8,1,2,4,12,6,18,9,3,15,30,10,5,35,7,21,42,14,28,56,8,
%T 16,48,24,72,36,2,1,3,9,18,6,12,24,8,40,20,10,5,55,11,33,66,22,44,4,
%U 52,26,78,39,13,91,7,49,98,14,42,126,63,21,105,15,45,135,27,81
%N Table T(n,k), n>=1, k>=1, row n being the lexicographically earliest of the longest sequences of distinct positive integers in which the k-th term does not exceed n*k and the smaller of adjacent terms divides the larger, giving a prime.
%C The longest sequence is finite for all n. We can deduce this, because we know from the work of Saias that A337125(m)/m * log m is bounded, where A337125(m) is the length of the longest simple path in the divisor graph of {1,...,m}. See the comment in A337125 giving constraints on its terms.
%C The sequence of row lengths starts 2, 6, 25, 97.
%H Peter Munn, <a href="/A340114/b340114.txt">Rows n = 1..4 of triangle, flattened</a>
%H E. Saias, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa83/aa8333.pdf">Applications des entiers à diviseurs denses</a>, Acta Arithmetica, 83, 3 (1998), 225-240.
%F For n >= 1, 1 <= k <= row length(n), T(n,k) <= n * k.
%F For n >= 1, 1 <= k < row length(n), max(T(n,k+1)/T(n,k), T(n,k)/T(n,k+1)) is in A000040.
%e For n = 1, the only sequences of distinct positive integers that have their k-th term not exceeding 1*k = k, are those whose n-th term is k. The longest such sequence in which the smaller of adjacent terms divides the larger, giving a prime, is (1, 2), since 3/2 is 1.5. So row 1 has length 2, with T(1,1) = 1, T(1,2) = 2.
%e Table begins:
%e 1, 2;
%e 1, 3, 6, 2, 4, 8;
%e 1, 2, 4, 12, 6, 18, 9, 3, 15, 30, 10, 5, 35, 7, 21, 42, 14, 28, 56, 8, 16, 48, 24, 72, 36;
%e ...
%Y Cf. A000040, A300012, A337125, A339491.
%K nonn,hard,tabf
%O 1,2
%A _Peter Munn_, Dec 28 2020
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