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A214492 Arithmetic mean of next a(n) successive squares of positive integers is a square. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Only squares of positive integers, starting from 1, zero is not included (if it was included: A103214).

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

LINKS

Table of n, a(n) for n=1..46.

EXAMPLE

(1+4+9+...+337^2)/337 = 38025, because 38025 is a perfect square, a(1)=337;

(338^2+...+(338+648)^2)/649 = 473344, because 473344 is a perfect square, a(2)=649.

PROG

(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*k<sum:

        sqr+=1

    while sqr*sqr*k>sum:

        sqr-=1

    if sqr*sqr*k==sum and k>1:

        print k,

        sum = k = 0

CROSSREFS

Cf. A000290, A073684.

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

Sequence in context: A020358 A260540 A051962 * A157999 A152853 A142830

Adjacent sequences:  A214489 A214490 A214491 * A214493 A214494 A214495

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, Jul 19 2012

STATUS

approved

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Last modified May 7 03:48 EDT 2021. Contains 343636 sequences. (Running on oeis4.)