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
Keywords
Examples
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.
Links
- Harry J. Smith, Table of n, a(n) for n = 0..500
Programs
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Mathematica
Do[s = n^2; k = 1; While[s = s + k^2; !IntegerQ[ Sqrt[s]], k++ ]; Print[s], {n, 0, 30} ] -
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
Extensions
Edited by Jon E. Schoenfield, Jun 14 2018
Name clarified by Michel Marcus, Mar 24 2020
Comments