Bounded by cubes

$1-7$ divisible by $1$

$>2^3$ : All even from $8-26$ are divisible by both 1,2 and $<3^3$

$>3^3$ : All factors of 6 from $30-60$ are divisible by 1,2,3 and $<4^3$

$>4^3$ : All factors of 12 from $72-120$ are divisible by 1,2,3,4 and $<5^3$

$>5^3$ : Only $180$ is divisible by 1,2,3,4,5 and $<6^3$

$>6^3$ : All factors of 60 from $240-300$ are divisible by 1,2,3,4,5,6 and $<7^3$

$>7^3$ : only $420$ is divisible by 1,2,3,4,5,6,7 and $<8^3$

After this, there is $840$ is the next integer divisible by 1,2,3,4,5,6,7 but it is $>9^3$ and not divisible by 9

A possible approach:

assume numbers from $1$ to $y$ divide a number in the range from $y^3$ to $(y+1)^3$ . then

$y^3 \leq lcm\{i\}_{i=1}^{y} <(y+1)^3$

note that $lcm\{i\}_{i=1}^{y}$ is the least common multiplier fall the numbers form $1$ to $y$ .

also, we have

$lcm\{i\}_{i=1}^{y}= \sum_{p<y} p^{log_{p}y}$

trying a few values for $y$ suggests that $y \leq 7$ , in order for the very first inequality to hold true. then we take $lcm\{i\}_{i=1}^{7}=420$ , which is the answer.

$1^3<n<2^3$ we have numbers of the form $m$ . Greatest is $7$ .

$2^3<n<3^3$ we have numbers of the form $2m$ . Greatest is $26$ .

$3^3<n<4^3$ we have numbers of the form $6m$ . Greatest is $60$ .

$4^3<n<5^3$ we have numbers of the form $12m$ . Greatest is $120$ .

$5^3<n<6^3$ we have numbers of the form $60m$ . Greatest is $180$ .

$6^3<n<7^3$ we have numbers of the form $60m$ . Greatest is $300$ .

$7^3<n<8^3$ we have numbers of the form $420m$ . Greatest is $420$ .

$8^3<n<9^3$ we have numbers of the form $840m$ . Too high.

$9^3<n<10^3$ we have numbers of the form $2520m$ . Too high.

$10^3<n<11^3$ we have numbers of the form $2520m$ . Too high.

Claim , for $k^{th}$ prime $p_k$ greater than $3$ , $p_k^4 > p_{k+1}^3$ .

Hence, $11^3<n<12^3$ and $12^3<n<13^3$ we need a number $\geq 11*2520 > 11^4 > 13^3$ . Too high.

For $k>6$ , $p_k^3<n<p_{k+1}^3$ we need a number $\geq 11*13*...*p_k * 2520 > 11^4 *13*...*p_k > p_k^4 > p_{k+1}^3$ . Too high.

Hence max $n$ is $420$ .

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I believe the claim is true, but don't have a formal proof to hand.

Searching for successful small integers gives the first break in the sequence at 9. These gives an upper limit to be considered of the LCM of the positive integers from 1 to 8, which is 840. Searching the range from 1 to 840 gives 1,2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,26,30,36,42,48,54,60,72,84,96,108,120,180,240,300 and 420.