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A236247 Sequence of distinct least squares such that the arithmetic mean of the first n squares is also a square.

%I A236247 #19 May 20 2025 12:52:26
%S A236247 1,49,25,121,784,196,33124,4900,4,4356,2304324,213444,2371600,379456,
%T A236247 87616,360000,3802500,562500,100,532900,5456896,767376,5934096,992016,
%U A236247 9947716,1350244,32467204,44100,2414916,10458756,2683044
%N A236247 Sequence of distinct least squares such that the arithmetic mean of the first n squares is also a square.
%H A236247 Giovanni Resta, <a href="/A236247/b236247.txt">Table of n, a(n) for n = 1..200</a>
%F A236247 a(n) = A141391(n)^2
%e A236247 a(1) = 1.
%e A236247 a(2) is the smallest unused square such that (a(2)+a(1))/2 is a square. So, a(2) = 49.
%e A236247 a(3) is the smallest unused square such that (a(3)+a(2)+a(1))/3 is a square. So, a(3) = 25.
%e A236247 ...and so on.
%o A236247 (Python)
%o A236247 def Sq(x):
%o A236247   for n in range(10**15):
%o A236247     if x == n**2:
%o A236247       return True
%o A236247     if x < n**2:
%o A236247       return False
%o A236247   return False
%o A236247 def SqAve(init):
%o A236247   print(init)
%o A236247   lst = []
%o A236247   lst.append(init)
%o A236247   n = 1
%o A236247   while n < 10**9:
%o A236247     if n**2 not in lst:
%o A236247       if Sq(((sum(lst)+n**2)/(len(lst)+1))):
%o A236247         print(n**2)
%o A236247         lst.append(n**2)
%o A236247         n = 1
%o A236247       else:
%o A236247         n += 1
%o A236247     else:
%o A236247       n += 1
%o A236247 SqAve(1)
%Y A236247 Cf. A019444, A141391.
%K A236247 nonn
%O A236247 1,2
%A A236247 _Derek Orr_, Jan 20 2014