A068869 -id:A068869 - cp's OEIS frontend

A181908 Least k such that log(ceiling(sqrt(k!))^2-k!)/k > n.

17, 145, 1095, 8098, 59868, 442406, 3269011

Offset: 1

Artur Jasinski, Apr 01 2012

This sequence show how quickly A068869 increase in a logarithmic scale.

			a(1)=17 because log(ceiling(sqrt(17!))^2-17!)/17 = 1.00471 > 1.

Cf. A068869.

kkk = 0; n = 1; Do[While[kkk < max, n++; kk = Floor[Sqrt[n!]]; kkk = N[Log[(kk + 1)^2 - n!]/n]]; Print[n], {max, 1, 5}]

a(n)=my(k=solve(x=1,4<<(3*n),(log(2)+lngamma(x+1)/2)/x-n)\1,f=k!);while(n>log((sqrtint(f*=k++)+1)^2-f)/k,);k \\ Charles R Greathouse IV, Apr 03 2012

a(6)-a(7) from Charles R Greathouse IV, Apr 03 2012

A232802 Number of solution pairs (x,y) for x <= 11 such that x! + n = y^2 (Brocard-Ramanujan Diophantine equation) is soluble over the integers.

Original entry on oeis.org

3, 1, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 1, 1, 0

Offset: 1

Frank M Jackson, Nov 30 2013

nonn

The Mathematica program will find the number of integer pairs (x,y) solving x!+n = y^2 for each n from 1 to 200 with x not exceeding 11. Dabrowski showed that the abc conjecture implies only finite solutions for each n. Berndt and Galway found that 11 was the highest value that x reached for a solution with n in the range 1 to 2500 and could find no further solution pairs (x,y) in that range even when x was increased to 10^5.

For n = 1 the number of solutions and arbitrary x is Brocard's problem, and it is conjectured - but verified only in the range x <= 10^12 - that there are 3 solution pairs (x,y): (4,5), (5,11), (7,71). - Georg Fischer, Nov 27 2020

Bruce Berndt and William Galway, On the Diophantine equation n! + 1 = m^2
Andrzej Dabrowski, On the Diophantine equation x! + A = y^2, Nieuw Archief voor Wiskunde, Vierde serie Deel 14 No. 3 (Nov. 1996) pp. 321-324.
Wikipedia, Brocard's problem
Wikipedia, abc conjecture - some consequences

Cf. A085692, A146968, A216071 (Brocard's problem; all essentially the same sequence).

Cf. A038202, A068869.

Table[Length@Select[Sqrt[Range[11]!+n], IntegerQ[#] &], {n, 1, 200}]

Definition narrowed by Georg Fischer, Nov 27 2020

cp's OEIS Frontend

A181908 Least k such that log(ceiling(sqrt(k!))^2-k!)/k > n.

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