A217551 -id:A217551 - cp's OEIS frontend

A217541 Smallest numbers n such that s! + n^2 and (s+1)! + n^2 are squares for some s.

1, 108, 108, 1140, 288, 35280, 1068480, 88361280, 4409475840, 2094434496000, 868006971127296000

Offset: 1

Robin Garcia, Oct 06 2012

The values of s are: 4, 8, 9, 10, 12, 14, 16, 18, 22, 24, 32.
It can be seen that n is, on average, an increasing function. (It is constant at s = 8 and s = 9 and decreases at s = 12). If proved this would show there is no repetition of a value of n for which simultaneously s! + n^2 = b^2 and (s+k)! + n^2 = c^2 for general and large values of k (not only for k = 1) and would solve Brocard's Problem: Exactly, the only 3 solutions to s! + 1 = b^2 are (4,5); (5,11) and (7,71).
Note that n^2 was chosen a square, but this is not necessary.
More terms of the sequence are hard to get if the program based on a simple algorithm, needing 10^9 bytes memory, is not improved in the sense of reducing the number of divisors used. This could probably be done.

			4! + 1 = 5^2 and 5! + 1 = 11^2.
8! + 108^2 = 228^2 and 9! + 108^2 = 612^2.
9! + 108^2 = 612^2 and 10! + 108^2 = 1908^2.
10! + 1140^2 = 2220^2 and 11! + 1140^2 = 6420^2.

Cf. A217277, A217541, A217550, A217551, A217553.

for(n=4,34,a=n!;b=n*a;s=sqrtint(a)+1+sqrtint((n+1)*a)+1;c=divisors(b);for(i=2,#c-1,if(s<=c[i],s=c[i];r=b\s;if(r%2==1,s=c[i+1]);r=b/s;d=(s-r)/2;t=d^2-a;if(issquare(t),print1(sqrtint(t),",  ");next(2)))))

A217550 Numbers n such that s! + n^2 and (s + 2)! + n^2 are squares for some s, ordered by s.

Original entry on oeis.org

1, 179, 204, 108, 996, 2934, 81720, 2152080, 851040, 271106640, 7935621120, 1143137318400, 52250931532800

Offset: 1

Robin Garcia, Oct 06 2012

nonn

The corresponding numbers s are: 5, 6, 7, 8, 9, 11, 12, 14, 16, 17, 19, 25, 27.

			5! + 1 = 11^2 and 7! + 1 = 71^2.
6! + 179^2 = 181^2 and 8! + 179^2 = 269^2.
7! + 204^2 = 216^2 and 9! + 204^2 = 636^2.
8! + 108^2 = 228^2 and 10! + 108^2 = 1908^2.

Cf. A217277, A217541, A217550, A217551, A217553.

for(n=4,32,a=n!;b=((n+2)*(n+1)-1)*a;c=divisors(b);for(i=2,#c-1,s=c[i];r=b\s;if(r

cp's OEIS Frontend

A217541 Smallest numbers n such that s! + n^2 and (s+1)! + n^2 are squares for some s.

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