A388302 a(n) = is the smallest m >= 1 such that n! = b_1*...*b_t with b_1 < ... < b_t and m = b_t - b_1.

1, 1, 2, 2, 2, 2, 4, 6, 7, 6, 9, 9, 9, 12, 14, 12, 15, 16, 17, 18, 19, 19, 20, 21, 22, 24, 24, 24, 27, 27, 29, 30, 30, 32, 34, 33, 34, 34, 37, 37, 38, 38, 39, 41, 41, 41, 44, 45, 46, 46, 48, 48, 50, 50, 52, 53, 53, 53, 56, 56, 56, 58, 62, 62, 64, 64, 64, 64, 66, 66, 68, 68, 68, 70, 71, 72, 73, 72

Offset

2,3

Comments

Empirical observation: while generating this sequence, it seems like searching for factors around a window of (n!)^(1/k), where 2 <= k <= n will find likely matches. This is due to the geometric mean of k factors being near (n!)^(1/k). No proof if this heuristic method guarantees finding valid factors, or a numerical lower bound on the window width.

From David A. Corneth, Sep 17 2025: (Start)

When m = 0 is allowed we would have a(n) = 0 for all n and we could define a(1).

As m >= 1, t > 1 and so b_1*b_t <= n! and so b_1 < sqrt(n!) (< as for n >= 2 we have n! is never a square).

More generally b_1 < (n!)^(1/t).

a(n) <= min(A061057(n), n-2) for n>=3 where upper bound n-2 comes from t = n-1, b_1 = 2, b_t = n. (End)

References

P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980), p. 76.

Links

Table of n, a(n) for n=2..79.

Thomas Bloom, Problem 393, Erdős Problems.

David A. Corneth, PARI program

David A. Corneth, List of tuples multiplying to n! and giving a(n) for n = 2..79

Charlie Liou, Python program using geometric mean heuristic

Erdős problems database contributors, Issue 92 linking Erdős problems to the OEIS.

Example

For n = 3, a(3) = 1 because 3! = 2*3 and 3 - 2 = 1.
For n = 8, a(8) = 4 with two solutions:
  8! = 32*35*36, 36-32 = 4
  8! = 12*14*15*16, 16-12 = 4.

Mathematica

gens[m_, x_]:=Join[{x}, #, {x+m}]&/@Subsets[x+Range[1, m-1]]
decode[s_, t_]:=If[Length[t]==0, t, s/.t]
process[nf_, s_, x_]:=decode[s, Solve[Times@@s==nf, x, Integers]]
do[n_, m_]:=Flatten[process[n!, #, x]&/@gens[m, x], 1]
do[n_]:=Module[{m, d}, For[m=1, True, m++, d=do[n, m]; If[Length[d]>0, Return[d]]]]
(* From Boris Alexeev (Erdős problems database contributor to Issue 92) who computed a(21)-a(22) *)

Prog

(Python)
from math import prod, factorial
from sympy.ntheory import factorint
from sympy.utilities.iterables import multiset_partitions
def a(n):
    prime_factors = [factors for prime in [[i for _ in range(j)] for i, j in factorint(factorial(n)).items()] for factors in prime]
    best_m = float('inf')
    for partition in multiset_partitions(prime_factors):
        factors = sorted([prod(group) for group in partition])
        if len(factors) != len(set(factors)): continue
        test_m = factors[-1] - factors[0]
        if test_m < best_m and len(factors) >= 2: best_m = test_m
    return best_m
print([a(n) for n in range(3, 13)])
(PARI) \\ See also David A. Corneth link
R(r, m, u)={my(t=r, e=logint(r, m+1)); if(e>1 && sqrtnint(r, e)<u, fordiv(r, d, my(d2=r/d); if(d>d2 && d2>m && u>d2, t=min(t, self()(d, d2, u))))); t}
a(n)={my(r=n!, t=r-1); fordiv(r, d, my(d2=r/d); if(d>d2, t=min(t, R(d, d2, d2+t)-d2))); t} \\ Andrew Howroyd, Sep 16 2025

Crossrefs

Cf. A061057, A082119.

Sequence in context: A008363 A261611 A097198 * A126663 A032600 A103260

Adjacent sequences: A388299 A388300 A388301 * A388303 A388304 A388305

Keyword

nonn

Author

Charlie Liou, Sep 16 2025

Extensions

a(23)-a(40) from Andrew Howroyd, Sep 16 2025

a(41)-a(79) from David A. Corneth, Sep 19 2025

Status

approved