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A052261
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Smallest integer that can be expressed as the sum of n squares of positive integers in exactly n distinct ways, or 0 if no such number exists.
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2
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1, 50, 54, 52, 53, 54, 55, 56, 57, 61, 67, 66, 67, 68, 74, 0, 79, 83, 87, 83, 84, 88, 0, 93, 96, 105, 101, 110, 106, 102, 116, 0, 108, 0, 0, 0, 117, 0, 117, 121, 0, 125, 0, 135, 0, 0, 0, 134, 0, 137, 145, 144, 143, 0, 0, 156, 0, 0, 152, 0, 0, 157, 0, 0, 0, 169, 0, 166, 0, 166, 0
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OFFSET
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1,2
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COMMENTS
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a(16) > 119; values for a(17) through a(22) are 79, 83, 87, 83, 84, 88.
a(16) > 10000, a(23) > 10000, if they exist. - Naohiro Nomoto, Aug 22 2001
If the number of ways to write m as the sum of n squares is at least x for m in the range k^2 to 2*k^2 + 2*k + 1, it is at least x for any larger m; take the smallest square j^2 greater than x/2 and x - j^2 >= m^2 has at least x representations, none of which obviously can exceed j^2. The 0's for n=16 and 23 can be verified in this way with k=10; 32 to 35 with k=11; 36 with k=12; 38 to 49 with k=13; and 54 up with k=14. For n sufficiently large, a(n) = (index of n in A111178) + n, or 0 if n does not occur in A111178. - Franklin T. Adams-Watters, Jul 15 2006
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LINKS
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EXAMPLE
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a(2) = 50 = 1^2 + 7^2 = 5^2 + 5^2.
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MATHEMATICA
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a[1] = b[1] = 1; a[2] = b[2] = 50; b[n_] := b[n] = If[a[n-1] > 0, a[n-1], b[n-1]]; a[n_] := a[n] = (an=0; For[k = b[n-1]-8, k <= b[n-1]+14, k++, pr = PowersRepresentations[k, n, 2]; If[n == Count[pr, r_ /; FreeQ[r, 0]], an = k; Break[]]]; an); Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 71}](* Jean-François Alcover, Jan 27 2012 *)
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PROG
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(PARI) numsumsq(n, m) = local(p, i); p=1+x*O(x^m)+y*O(y^n); for(i=1, sqrtint(m), p=p/(1-x^i^2*y)); p=polcoeff(p, n, y); vector(m, i, polcoeff(p, i))
vecfind(v, x) = local(i, y); for(i=1, matsize(v)[2], if(v[i]==x, y=i; break())); y
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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David M. Grumm (dmg(AT)head-cfa.harvard.edu), Feb 03 2000
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EXTENSIONS
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STATUS
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approved
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