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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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..91.

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).

Cf. A339491, A339496, A339489.

Sequence in context: A230296 A278317 A086454 * A069360 A175509 A213023

Adjacent sequences:  A339489 A339490 A339491 * A339493 A339494 A339495

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Dec 31 2020

STATUS

approved

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Last modified May 8 13:26 EDT 2021. Contains 343666 sequences. (Running on oeis4.)