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A339492
T(n, k) = tau(k) + floor(n/k) - 1, where tau = A000005. Triangle read by rows.
3
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, 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, 7, 5, 5, 3, 5, 2, 4, 3, 4, 2, 6, 13, 7, 5, 5, 3, 5, 2, 4, 3, 4, 2, 6, 2
OFFSET
1,2
COMMENTS
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.
FORMULA
T(n, k) = card(divisors(k) union {k*j : j = 2..floor(n/k)}).
EXAMPLE
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}.
The triangle starts:
[1] 1;
[2] 2, 2;
[3] 3, 2, 2;
[4] 4, 3, 2, 3;
[5] 5, 3, 2, 3, 2;
[6] 6, 4, 3, 3, 2, 4;
[7] 7, 4, 3, 3, 2, 4, 2;
[8] 8, 5, 3, 4, 2, 4, 2, 4;
[9] 9, 5, 4, 4, 2, 4, 2, 4, 3;
[10] 10, 6, 4, 4, 3, 4, 2, 4, 3, 4.
MAPLE
T := (n, k) -> NumberTheory:-tau(k) + iquo(n, k) - 1:
seq(seq(T(n, k), k = 1..n), n = 1..13);
CROSSREFS
T(n, 1) = A000027(n), T(n, n) = A000005(n), T(2n, n) = A334954(n).
Sequence in context: A230296 A278317 A086454 * A069360 A175509 A213023
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 31 2020
STATUS
approved