cp's OEIS Frontend

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.

%I A052261 #27 Dec 22 2021 02:55:08
%S A052261 1,50,54,52,53,54,55,56,57,61,67,66,67,68,74,0,79,83,87,83,84,88,0,93,
%T A052261 96,105,101,110,106,102,116,0,108,0,0,0,117,0,117,121,0,125,0,135,0,0,
%U A052261 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
%N 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.
%C A052261 a(16) > 119; values for a(17) through a(22) are 79, 83, 87, 83, 84, 88.
%C A052261 a(16) > 10000, a(23) > 10000, if they exist. - _Naohiro Nomoto_, Aug 22 2001
%C A052261 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
%H A052261 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A052261 a(2) = 50 = 1^2 + 7^2 = 5^2 + 5^2.
%t A052261 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 *)
%o A052261 (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))
%o A052261 vecfind(v, x) = local(i, y); for(i=1,matsize(v)[2],if(v[i]==x,y=i;break()));y
%o A052261 a(n, m=1000) = v=numsumsq(n,m);vecfind(v,n) /* values must be checked */ \\ _Franklin T. Adams-Watters_, Jul 15 2006
%Y A052261 Cf. A111178, A215908.
%K A052261 nonn,nice
%O A052261 1,2
%A A052261 David M. Grumm (dmg(AT)head-cfa.harvard.edu), Feb 03 2000
%E A052261 More terms from _Franklin T. Adams-Watters_, Jul 15 2006