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A260375
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Numbers k such that A260374(k) is a perfect square.
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1
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0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 14, 15, 16
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OFFSET
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1,3
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COMMENTS
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There are a surprising number of small terms in this sequence.
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
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LINKS
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EXAMPLE
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6! = 720. The nearest perfect square is 729. The difference is 9, which is itself a perfect square. So, 6 is in this sequence.
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PROG
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(Python)
from gmpy2 import isqrt, is_square
for i in range(1, 1001):
g *= i
s = isqrt(g)
t = g-s**2
if is_square(t if t-s <= 0 else 2*s+1-t):
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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