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%I #17 Dec 31 2020 13:13:22
%S 1,2,2,3,2,2,4,3,2,3,5,3,2,3,2,6,4,3,3,2,4,7,4,3,3,2,4,2,8,5,3,4,2,4,
%T 2,4,9,5,4,4,2,4,2,4,3,10,6,4,4,3,4,2,4,3,4,11,6,4,4,3,4,2,4,3,4,2,12,
%U 7,5,5,3,5,2,4,3,4,2,6,13,7,5,5,3,5,2,4,3,4,2,6,2
%N T(n, k) = tau(k) + floor(n/k) - 1, where tau = A000005. Triangle read by rows.
%C A simple path in the divisor graph of {1,...,n} is a sequence of distinct numbers between 1 and n such that if m immediately follows k, then either m divides k or k divides m. Let S(n, k) = divisors(k) union {k*j : j = 2..floor(n/k)}. A path p is only valid if the elements of the path p(k-1) are in S(n, p(k)), for k = 2..n.
%F T(n, k) = card(divisors(k) union {k*j : j = 2..floor(n/k)}).
%e Row 6 lists the cardinalities of the sets {1, 2, 3, 4, 5, 6}, {1, 2, 4, 6}, {1, 3, 6}, {1, 2, 4}, {1, 5}, {1, 2, 3, 6}.
%e The triangle starts:
%e [1] 1;
%e [2] 2, 2;
%e [3] 3, 2, 2;
%e [4] 4, 3, 2, 3;
%e [5] 5, 3, 2, 3, 2;
%e [6] 6, 4, 3, 3, 2, 4;
%e [7] 7, 4, 3, 3, 2, 4, 2;
%e [8] 8, 5, 3, 4, 2, 4, 2, 4;
%e [9] 9, 5, 4, 4, 2, 4, 2, 4, 3;
%e [10] 10, 6, 4, 4, 3, 4, 2, 4, 3, 4.
%p T := (n, k) -> NumberTheory:-tau(k) + iquo(n, k) - 1:
%p seq(seq(T(n, k), k = 1..n), n = 1..13);
%Y T(n, 1) = A000027(n), T(n, n) = A000005(n), T(2n, n) = A334954(n).
%Y Cf. A339491, A339496, A339489.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Dec 31 2020
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