A055228 -id:A055228 - cp's OEIS frontend

A181896 Least value x solving x^2 - y^2 = n!

5, 11, 27, 71, 201, 603, 1905, 6318, 21888, 78912, 295260, 1143536, 4574144, 18859680, 80014848, 348776640, 1559776320, 7147792848, 33526120320, 160785625902, 787685472000, 3938427360000, 20082117976800, 104349745817240, 552166953609600, 2973510046027938

Offset: 4

Artur Jasinski, Mar 31 2012

nonn

Many of terms in this sequence are that same as A055228(n) but not all.

a(n) solves the Brocard-Ramanujan Problem, n! = a(n)^2 - 1, and thus (n, a(n)) are a pair of Brown Numbers, if and only if A038202(n) = 1. - Austin Hinkel, Dec 28 2022

Eric Weisstein's World of Mathematics, Brocard's Problem.

For least y value see A038202.

Cf. A055228.

cc = {}; Do[f = n!/4; x = Max[Select[Divisors[f], # <= Sqrt[f] &]]; kk = f/x - x; AppendTo[cc, Sqrt[n! + kk^2]], {n, 4, 30}]; cc

a(n)=my(N=n!,x=sqrtint(N));while(!issquare(x++^2-N),);x \\ Charles R Greathouse IV, Apr 10 2012

A273932 Integers m such that ceiling(sqrt(m!)) is prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 21, 2132, 3084, 9301

Offset: 1

Salvador Cerdá, Jun 04 2016

This sequence includes the known solutions of Brocard's problem as of 2016 (see A146968).

			3 is a term because 3! = 6, sqrt(6) = 2.449489742783178..., the ceiling of which is 3, which is prime.
4 is a term because 4! = 24, sqrt(24) = 4.898979485566356..., the ceiling of which is 5, which is prime.

Cf. A055228 (ceiling(sqrt(n!))), A146968.

Select[Range[3200], PrimeQ[Ceiling[Sqrt[#!]]] &]

from math import isqrt, factorial
from itertools import count, islice
from sympy import isprime
def A273932_gen(): # generator of terms
    return filter(lambda n:isprime(1+isqrt(factorial(n)-1)),count(1))
A273932_list = list(islice(A273932_gen(),7)) # Chai Wah Wu, Jul 29 2022

a(9) from Giovanni Resta, Jun 20 2016

cp's OEIS Frontend

A181896 Least value x solving x^2 - y^2 = n!

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