OFFSET
1,1
COMMENTS
Has a relatively large intersection with A084920, but neither contains nor is contained in that sequence.
It is conjectured that this sequence is asymptotically sparser than A070003 (Erdős problem #380).
From Michael De Vlieger, Sep 23 2025: (Start)
Since A001694 \ {1} is a proper subset of A070003 and since 1 is not in A388654, there are no powerful numbers in this sequence.
Since squarefree numbers k (in A005117) are such that gpf(k)^2 does not divide k, thus are not in A070003, those squarefree numbers in A388654 also appear in this sequence.
This sequence is a proper subset of A052485.
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Thomas Bloom, Problem 380, Erdős Problems.
Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4 (1976), pp. 337-355.
Paul Erdős and Ron L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980).
Terence Tao, Products of consecutive integers with unusual anatomy, arXiv:2603.27990 [math.NT], 2026. See p. 4 (Lemma 1.6).
EXAMPLE
1683 = 41^2+2 lies in this sequence, because [1681,1683] is a bad interval (the product is divisible by the square of its largest prime factor 41), but 1683=3^2*11*17 is not divisible the square of its largest prime factor 17.
MATHEMATICA
nn = 12000; s = Array[FactorInteger[#][[-1, 1]] &, nn + Floor@ Log2[nn]]; k = 1; Complement @@ Reverse@ Map[Union @* Flatten, TakeDrop[#, 1]] &@ 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 23 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Terence Tao, Sep 20 2025
EXTENSIONS
More terms from Pontus von Brömssen, Sep 20 2025
STATUS
approved