A052261 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.

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

Offset: 1

Author

David M. Grumm (dmg(AT)head-cfa.harvard.edu), Feb 03 2000

Keywords

• nonn
• nice

Comments

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

Examples

			a(2) = 50 = 1^2 + 7^2 = 5^2 + 5^2.
		

Links

• Index entries for sequences related to sums of squares

Crossrefs

Cf. A111178, A215908.

Programs

Mathematica

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 *)

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
a(n, m=1000) = v=numsumsq(n,m);vecfind(v,n) /* values must be checked */ \\ Franklin T. Adams-Watters, Jul 15 2006

Extensions

More terms from Franklin T. Adams-Watters, Jul 15 2006