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A214492 Arithmetic mean of next a(n) successive squares of positive integers is a square.

%I A214492 #19 May 22 2021 20:57:53
%S A214492 337,649,961,1273,1585,1897,1919,1151,1223,1295,1367,1439,1511,1583,
%T A214492 1655,1727,1799,1871,1943,2015,2087,2159,2231,2303,2375,2447,2519,
%U A214492 1487,1511,1535,1559,1583,1607,1631,1655,1679,1703,1727,1751,1775,1799,1823,1847,1871,1895,1919
%N A214492 Arithmetic mean of next a(n) successive squares of positive integers is a square.
%C A214492 Only squares of positive integers, starting from 1; zero is not included. (If it were included, A103214 would result.)
%C A214492 Also, a(n)=1 is obviously not permitted: must be 2 or more successive squares, otherwise all a(n)=1.
%C A214492 Among first 1363 terms all are odd, 933 are primes, a(n) < a(n-1) twice.
%C A214492 Corresponding arithmetic means that are perfect squares:
%C A214492 b(n) = 38025, 473344, 2229049, 6812100, 16313521, 33408400, 59013124, 84695209, 107952100, 135699201, 168480400, 206870689, 251476164, 302934025, 361912576, 429111225, 505260484, 591121969, 687488400
%C A214492 Their square roots c(n) = sqrt(b(n)):
%C A214492 195, 688, 1493, 2610, 4039, 5780, 7682, 9203, 10390, 11649, 12980, 14383, 15858, 17405, 19024, 20715, 22478, 24313, 26220, 28199, 30250, 32373, 34568, 36835, 39174, 41585, 44068, 46067, 47566, 49089, 50636, 52207
%e A214492 (1 + 4 + 9 + ... + 337^2)/337 = 38025, which is a square, so 337 is a term.
%e A214492 (338^2 + ... + (338 + 648)^2)/649 = 473344, which is a square, so 649 is a term.
%o A214492 (Python)
%o A214492 import math
%o A214492 sum = k = 0
%o A214492 for n in range(1, 220000):
%o A214492     sum += n*n
%o A214492     k += 1
%o A214492     sqr = int(math.sqrt(sum*1.0/k))
%o A214492     while sqr*sqr*k<sum:
%o A214492         sqr+=1
%o A214492     while sqr*sqr*k>sum:
%o A214492         sqr-=1
%o A214492     if sqr*sqr*k==sum and k>1:
%o A214492         print(k, end=',')
%o A214492         sum = k = 0
%Y A214492 Cf. A000290, A073684.
%Y A214492 Cf. A103214, excluding first term: arithmetic mean of next a(n) successive squares of nonnegative integers is a square.
%K A214492 nonn
%O A214492 1,1
%A A214492 _Alex Ratushnyak_, Jul 19 2012