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A065611
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Least k such that n^2 + Sum (from 1 to k) m^2 is a perfect square.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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).
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,500
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EXAMPLE
| n = 3: a[4] = 64 because n^2+1+4+9+16+25 = 9+(1+4+9+16+25) = 64 = 8.8; n = 4: a[4] = 150700176 because n^2+(1+4+...+767^2) = 150700176 = 12276.12726, where 767 is the length of shortest such consecutive-square sequence which provides[when summed] a new square, namely 12276^2. Often the least solution is rather large. E.g. at n = 93, a[n] = 23850559947150225 which means that 93^2+A000330[415151] = 8649+ [a long square sum] = 154436265^2 = 23850559947150225.
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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} ]
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PROG
| (PARI) { for (n = 0, 500, s=n^2 + 1; k=1; while (!issquare(s), k++; s+=k^2); write("b065611.txt", n, " ", s) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Oct 23 2009]
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CROSSREFS
| Cf. A000330, A065610-A065615.
Sequence in context: A172712 A161023 A156405 * A101252 A133281 A034605
Adjacent sequences: A065608 A065609 A065610 * A065612 A065613 A065614
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Nov 07 2001
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