A065611 Let k be the least integer such that n^2 + Sum_{m=1..k} m^2 is a perfect square, then a(n) is the resulting square.

1, 35721, 9, 64, 150700176, 1521, 1718434116, 3844, 1849, 900, 2209, 474721, 529, 116964, 400, 419845682025, 618399795456, 3600, 187489, 1734149230641, 10816, 1681, 5560164, 2025, 961, 1444, 961, 784, 41209, 21926752125201

Offset

0,2

Comments

I.e., n^2 + {1 + 4 + 9 + 16 + ... + m^2} = a(n) = A065612(n)^2 = A065311(n). a(n) is the smallest square obtained as n^2 + x*(x+1)*(2x+1)/6 where x = A065610(n).

Links

Harry J. Smith, Table of n, a(n) for n = 0..500

Example

n = 3: a(3) = 64 because n^2 + 1 + 4 + 9 + 16 + 25 = 9 + (1 + 4 + 9 + 16 + 25) = 64 = 8^2;
n = 4: a(4) = 150700176 because n^2 + (1 + 4 + ... + 767^2) = 150700176 = 12276^2, where 767 is the length of the shortest such consecutive-square sequence which provides(when summed) a new square, namely 12276^2. Often the least solution is rather large. E.g., a(93) = 23850559947150225 which means that 93^2 + A000330(415151) = 8649 + [a long square sum] = 154436265^2 = 23850559947150225.

Mathematica

Do[s = n^2; k = 1; While[s = s + k^2; !IntegerQ[ Sqrt[s]], k++ ]; Print[s], {n, 0, 30} ]

Prog

(PARI) { for (n = 0, 500, s=n^2 + 1; k=1; while (!issquare(s), k++; s+=k^2); write("b065611.txt", n, " ", s) ) } \\ Harry J. Smith, Oct 23 2009
(PARI) a(n) = my(s=n^2+1, k=1); while (!issquare(s), k++; s+=k^2); s; \\ Michel Marcus, Mar 24 2020

Crossrefs

Cf. A000330, A065610, A065612.

Sequence in context: A161023 A252920 A156405 * A101252 A133281 A326320

Adjacent sequences: A065608 A065609 A065610 * A065612 A065613 A065614

Keyword

nonn

Author

Labos Elemer, Nov 07 2001

Extensions

Edited by Jon E. Schoenfield, Jun 14 2018

Name clarified by Michel Marcus, Mar 24 2020

Status

approved