A214492 - OEIS

%I #19 May 22 2021 20:57:53

%S 337,649,961,1273,1585,1897,1919,1151,1223,1295,1367,1439,1511,1583,

%T 1655,1727,1799,1871,1943,2015,2087,2159,2231,2303,2375,2447,2519,

%U 1487,1511,1535,1559,1583,1607,1631,1655,1679,1703,1727,1751,1775,1799,1823,1847,1871,1895,1919

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

%C Only squares of positive integers, starting from 1; zero is not included. (If it were included, A103214 would result.)

%C Also, a(n)=1 is obviously not permitted: must be 2 or more successive squares, otherwise all a(n)=1.

%C Among first 1363 terms all are odd, 933 are primes, a(n) < a(n-1) twice.

%C Corresponding arithmetic means that are perfect squares:

%C b(n) = 38025, 473344, 2229049, 6812100, 16313521, 33408400, 59013124, 84695209, 107952100, 135699201, 168480400, 206870689, 251476164, 302934025, 361912576, 429111225, 505260484, 591121969, 687488400

%C Their square roots c(n) = sqrt(b(n)):

%C 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 (1 + 4 + 9 + ... + 337^2)/337 = 38025, which is a square, so 337 is a term.

%e (338^2 + ... + (338 + 648)^2)/649 = 473344, which is a square, so 649 is a term.

%o (Python)

%o import math

%o sum = k = 0

%o for n in range(1, 220000):

%o     sum += n*n

%o     k += 1

%o     sqr = int(math.sqrt(sum*1.0/k))

%o     while sqr*sqr*k<sum:

%o         sqr+=1

%o     while sqr*sqr*k>sum:

%o         sqr-=1

%o     if sqr*sqr*k==sum and k>1:

%o         print(k, end=',')

%o         sum = k = 0

%Y Cf. A000290, A073684.

%Y Cf. A103214, excluding first term: arithmetic mean of next a(n) successive squares of nonnegative integers is a square.

%K nonn

%O 1,1

%A _Alex Ratushnyak_, Jul 19 2012