A087374 -id:A087374 - cp's OEIS frontend

A082995 Distance from n!+1 to next larger square.

2, 1, 2, 0, 0, 8, 0, 80, 728, 224, 323, 39168, 82943, 176399, 215295, 3444735, 26167683, 114349224, 255004928, 1158920360, 11638526760, 42128246888, 191052974115, 97216010328, 2430400258224, 1553580508515, 4666092737475

Offset: 1

Jason Earls, May 29 2003

nonn

The only known values of n such that n!+1 is a perfect square are 4, 5 and 7. Paul Leyland, et al. have found no other solutions for n <= 1 million (see link). For 1 <= n <= 11, n!+1 is within 1000 of being a square. Is there another n such that n!+1 <= "1000 away" from being a perfect square?

			a(5)=0 because 5!+1 is a square.
a(8)=80 because 8!+1 = 40321 and the next larger square is 40401, so 40401-40321 = 80.

Amiram Eldar, Table of n, a(n) for n = 1..808
P. Leyland, Solutions to n!+1=m^2.

Cf. A038507, A087374.

a[n_] := Ceiling @ Sqrt[(f = n! + 1)]^2 - f; Array[a, 27] (* Amiram Eldar, Dec 14 2019 *)

for(k=1,27,print1(ceil(sqrt(k!+1))^2-(k!+1),", ")) \\ Hugo Pfoertner, Dec 14 2019

A087375 Smallest n-th power > n!.

Original entry on oeis.org

2, 4, 8, 81, 243, 729, 16384, 65536, 1953125, 9765625, 48828125, 2176782336, 13060694016, 678223072849, 4747561509943, 33232930569601, 2251799813685248, 18014398509481984, 144115188075855872, 12157665459056928801, 109418989131512359209, 10000000000000000000000, 100000000000000000000000, 1000000000000000000000000, 108347059433883722041830251

Offset: 1

Amarnath Murthy, Sep 09 2003

nonn

			a(5) = 243 as 243 = 3^5 > 5! > 2^5.

Robert Israel, Table of n, a(n) for n = 1..448

Cf. A087374, A214046.

g:= proc(n)  floor(1+(n!)^(1/n))^n end proc:
map(g, [$1..30]); # Robert Israel, Nov 25 2024

a[n_]:=Floor[1+(n!)^(1/n)]^n; Array[a,25] (* Stefano Spezia, Mar 27 2025 *)

a(n) = A214046(n)^n for n > 1. - Robert Israel, Nov 25 2024

Corrected and extended by Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004

cp's OEIS Frontend

A082995 Distance from n!+1 to next larger square.

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