A387186 Numbers k such that m!*k is a perfect square for some m < k-1.

4, 6, 8, 9, 16, 18, 20, 24, 25, 28, 30, 32, 35, 36, 45, 49, 50, 54, 63, 64, 70, 72, 77, 80, 81, 96, 98, 100, 112, 120, 121, 125, 128, 140, 144, 150, 162, 169, 175, 180, 196, 200, 216, 225, 231, 242, 245, 252, 256, 270, 280, 288, 289, 294, 308, 315, 320, 324, 338

Offset

1,1

Comments

The condition that m!*k is a perfect square is equivalent to m!*(k-1)!*k! being a perfect square, or to A055204(m)=A007913(k).

Is a subsequence of A388851. Conjectured by Erdős and Graham to be the dominant component of that sequence (see also Erdős problem #374).

Contains A000290 (from 4 onwards), A001105 (from 8 onwards), A033429 (from 20 onwards), A033581 (from 6 onwards), etc. More generally, contains the sufficiently large square multiples of A055204(m) for each m.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Thomas Bloom, Problem 374, Erdős Problems.

P. Erdős and R. L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4 (1976), pp. 337-355.

P. Erdős and R. 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. 6 (Sect. 1.3).

Example

18 is a term because 2!18 = 6^2 is a perfect square.

Mathematica

Select[Range[300], AnyTrue[Range[# - 1]!*#, IntegerQ@ Sqrt[#] &] &] (* Michael De Vlieger, Sep 22 2025 *)

Prog

(Python)
import math
def is_square(n):
    r = math.isqrt(n)
    return r * r == n
results = []
for c in range(1, 301):
    for a in range(1, c-1):
        if is_square(math.factorial(a)*c):
            results.append((c, a))
            break
(Python)
from math import prod
from itertools import count, islice
from sympy import factorint
def A387186_gen(): # generator of terms
    g, r = set(), {1:1}
    for c in count(2):
        f = set(i for i, j in factorint(c).items() if j&1)
        g ^= f
        if (t:=prod(g)) not in r:
            r[t] = c
        e = prod(f)
        if e in r and r[e]<c-1:
            yield c
A387186_list = list(islice(A387186_gen(), 54)) # Chai Wah Wu, Sep 24 2025, updated Oct 05 2025

Crossrefs

Cf. A055204, A007913.

Subsequence of A388851.

Contains (most of) A000290, A001105, A033429, A033581.

Sequence in context: A228019 A228020 A163282 * A095404 A073540 A123710

Adjacent sequences: A387183 A387184 A387185 * A387187 A387188 A387189

Keyword

nonn

Author

Terence Tao, Sep 22 2025

Status

approved