cp's OEIS Frontend

A260375 Numbers k such that A260374(k) is a perfect square.

%I A260375 #21 Jul 18 2021 02:14:44
%S A260375 0,1,2,4,5,6,7,8,10,11,14,15,16
%N A260375 Numbers k such that A260374(k) is a perfect square.
%C A260375 There are a surprising number of small terms in this sequence.
%C A260375 Heuristic: The square root of x has an average distance of 1/4 to an integer, so |x - round(sqrt(x))^2| is around |x - (sqrt(x) - 1/4)^2| or about sqrt(x)/2, hence A260374(n) is around sqrt(n!)/2. By Stirling's approximation this is around (n/e)^(n/2) which is a square with probability (n/e)^(-n/4). The integral of this function converges, so this sequence should be finite. This heuristic is crude, though, because it does not model the extreme values of A260374. - _Charles R Greathouse IV_, Jul 23 2015
%C A260375 There are no further terms up to 10^5, so probably the list is complete. - _Charles R Greathouse IV_, Jul 23 2015
%e A260375 6! = 720. The nearest perfect square is 729. The difference is 9, which is itself a perfect square. So, 6 is in this sequence.
%o A260375 (PARI) is(n)=my(N=n!,s=sqrtint(N)); issquare(min(N-s^2, (s+1)^2-N)) \\ _Charles R Greathouse IV_, Jul 23 2015
%o A260375 (Python)
%o A260375 from gmpy2 import isqrt, is_square
%o A260375 A260375_list, g = [0], 1
%o A260375 for i in range(1, 1001):
%o A260375     g *= i
%o A260375     s = isqrt(g)
%o A260375     t = g-s**2
%o A260375     if is_square(t if t-s <= 0 else 2*s+1-t):
%o A260375         A260375_list.append(i) # _Chai Wah Wu_, Jul 23 2015
%Y A260375 Cf. A260373, A260374.
%K A260375 nonn,more
%O A260375 1,3
%A A260375 _Otis Tweneboah_, _Pratik Koirala_, _Eugene Fiorini_, _Nathan Fox_, Jul 23 2015