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Search: id:A065610
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| A065610 |
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Smallest number m so that n^2+A000330[m] is also a square, i.e. n^2 + {1+4+9+16+...+m^2}=w^2 for some w. |
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3
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| 1, 47, 2, 5, 767, 16, 1727, 22, 17, 13, 18, 112, 10, 70, 8, 10799, 12287, 21, 82, 17327, 31, 15, 255, 16, 10, 13, 9, 5, 49, 40367, 43199, 117, 17, 1630, 7, 58799, 10, 65711, 34, 73007, 49, 13, 64, 29, 17, 6, 9, 30, 42, 309, 8, 124847, 17, 31, 139967, 13, 150527, 15
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OFFSET
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0,2
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COMMENT
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I.e. a[n] is the least solution to n^2+[x(x+1)(2x+1)/6]=w^2; a[n] is the length of shortest sum of consecutive squares from 1 to a[n] which when added to n^2 gives a new square.
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FORMULA
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(n^2)+(1+4+9+.....+a[n]^2) = w^2, where w depends also on n; i.e. sum of consecutive squares from 1, 4, ...to a[n]^2 + n^2 is also a square.
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EXAMPLE
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n = 3: a[3] = 5 because n^2+1+4+9+16+25 = 9+(1+4+9+16+25) = 64 = 8.8; n = 4: a[4] = 767 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] = 415151, which means that 93^2+A000330[415151] = 8649+[long square sum] = 154436265^2 = 23850559947150225 is the smallest such square number, sum odd distinct consecutive squares except one repetition(8649)
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MATHEMATICA
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s=n^2 Do[s=s+m^2; If[IntegerQ[Sqrt[s]], Print[m]], {m, 1, 500000}] gives solutions of which the smallest is entered into the sequence.
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CROSSREFS
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Cf. A000330, A065311-A065315.
Sequence in context: A116328 A009038 A051319 this_sequence A033367 A052352 A089553
Adjacent sequences: A065607 A065608 A065609 this_sequence A065611 A065612 A065613
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KEYWORD
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nonn
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Nov 07 2001
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