A217541
Smallest numbers n such that s! + n^2 and (s+1)! + n^2 are squares for some s.
Original entry on oeis.org
1, 108, 108, 1140, 288, 35280, 1068480, 88361280, 4409475840, 2094434496000, 868006971127296000
Offset: 1
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.
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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
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.
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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
A217551
Smallest numbers n, for a given s, such that s! + n^2 and (s+3)! + n^2 are squares.
Original entry on oeis.org
1, 828, 508, 239499435, 4693095288000, 561589459200, 148245349824000
Offset: 1
4! + 1 = 5^2 and 7! + 1 = 71^2.
8! + 828^2 = 852^2 and 11! + 828^2 = 6372^2.
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for(n=4,30,a=n!;b=((n+3)*(n+2)*(n+1)-1)*a;c=divisors(b);for(i=2,#c-1,s=c[i];r=b\s;if(r
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