A389100 Elements of a non-singleton interval of natural numbers whose product is divisible by the square of the largest prime factor of the product.

8, 9, 24, 25, 48, 49, 50, 120, 121, 168, 169, 242, 243, 288, 289, 360, 361, 528, 529, 675, 676, 840, 841, 960, 961, 1155, 1156, 1368, 1369, 1444, 1445, 1680, 1681, 1682, 1683, 1848, 1849, 1850, 2208, 2209, 2210, 2400, 2401, 2600, 2601, 2645, 2646, 2736, 2737, 2738

Offset

1,1

Comments

All terms are composite.

The square of an odd prime and its predecessor are terms.

Conjecture: A387054 is a subsequence.

The conjecture above follows easily from the definitions: k is in A387054 if it is in A388654 but not in A070003. The former condition means that k is in an interval of numbers whose product is divisible by the square of the largest prime factor of the product, and the latter means that that interval cannot consist of k only. So k is a term here. - Pontus von Brömssen, Sep 26 2025

Links

Michael De Vlieger, Table of n, a(n) for n = 1..13396 (a(n) <= 2^25).

Thomas Bloom, Problem 380, Erdős Problems.

Peter Luschny, Python implementation.

Formula

This sequence U A070003 = A388654.

Example

1680, 1681, 1682, 1683 are terms because 41^2 divides their product and the product is not divisible by any prime larger than 41.

Mathematica

nn = 3000; s = Array[FactorInteger[#][[-1, 1]] &, nn + Floor@ Log2[nn]]; k = 2; Union@ Flatten@ Reap[While[Set[t, Select[Partition[Range[nn], k, 1], Divisible[#2, Max[s[[#1 ;; #1 + k - 1]]]^2] & @@ {First[#], Times @@ #} &]]; Length[t] > 0, Sow[t]; k++]][[-1, 1]] (* Michael De Vlieger, Sep 24 2025 *)

Prog

(Python)  # cf. links

Crossrefs

Cf. A070003, A388654, A387054.

Sequence in context: A304600 A278076 A394855 * A081351 A032462 A077599

Adjacent sequences: A389097 A389098 A389099 * A389101 A389102 A389103

Keyword

nonn

Author

Peter Luschny, Sep 23 2025

Status

approved