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

337, 649, 961, 1273, 1585, 1897, 1919, 1151, 1223, 1295, 1367, 1439, 1511, 1583, 1655, 1727, 1799, 1871, 1943, 2015, 2087, 2159, 2231, 2303, 2375, 2447, 2519, 1487, 1511, 1535, 1559, 1583, 1607, 1631, 1655, 1679, 1703, 1727, 1751, 1775, 1799, 1823, 1847, 1871, 1895, 1919

Offset: 1

Author

Alex Ratushnyak, Jul 19 2012

Keywords

• nonn

Comments

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

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

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

Corresponding arithmetic means that are perfect squares:

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

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

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

Examples

			(1 + 4 + 9 + ... + 337^2)/337 = 38025, which is a square, so 337 is a term.
(338^2 + ... + (338 + 648)^2)/649 = 473344, which is a square, so 649 is a term.
		

Crossrefs

Cf. A000290, A073684.

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

Programs

Python

import math
sum = k = 0
for n in range(1, 220000):
    sum += n*n
    k += 1
    sqr = int(math.sqrt(sum*1.0/k))
    while sqr*sqr*ksum:
        sqr-=1
    if sqr*sqr*k==sum and k>1:
        print(k, end=',')
        sum = k = 0