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A215537 Lowest k such that k is representable as both the sum of n and of n+1 nonzero squares.

%I A215537 #10 Aug 11 2015 01:13:01
%S A215537 25,17,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,
%T A215537 33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,
%U A215537 56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79
%N A215537 Lowest k such that k is representable as both the sum of n and of n+1 nonzero squares.
%e A215537 25 = 5^2 = 3^2 + 4^2
%e A215537 17 = 4^2 + 1^2 = 3^2 + 2^2 + 2^2
%e A215537 12 = 2^2 + 2^2 + 2^2 = 3^2 + 1^2 + 1^2 + 1^2
%e A215537 after this just add 1^2 to both sides.
%p A215537 # true if a is representable as a sum of n squares, each square >= m^2.
%p A215537 isRepnSqrsMin := proc(a,n,m)
%p A215537     local mpr ;
%p A215537     if a < n*m^2 then
%p A215537         return false;
%p A215537     end if;
%p A215537     if n = 1 then
%p A215537         if a>= m^2 and issqr(a) then
%p A215537             true;
%p A215537         else
%p A215537             false;
%p A215537         end if;
%p A215537     else
%p A215537         for mpr from m to a do
%p A215537             if a-mpr^2 < 1 then
%p A215537                 return false;
%p A215537             elif procname(a-mpr^2,n-1,mpr) then
%p A215537                 return true;
%p A215537             end if;
%p A215537         end do:
%p A215537     end if;
%p A215537 end proc:
%p A215537 # true if a is representable as a sum of n positive squares.
%p A215537 isRepnSqrs := proc(a,n)
%p A215537     isRepnSqrsMin(a,n,1) ;
%p A215537 end proc:
%p A215537 A215537 := proc(n)
%p A215537     local k;
%p A215537     for k from 1 do
%p A215537         if isRepnSqrs(k,n) and isRepnSqrs(k,n+1) then
%p A215537             return k;
%p A215537         end if;
%p A215537     end do:
%p A215537 end proc: # _R. J. Mathar_, Sep 11 2012
%Y A215537 Cf. A000290 (representable as sum of 1 square), A000404 (sum of 2 positive squares), A000408 (sum of 3 positive squares), A000414 (sum of 4 positive squares), A047700 (sum of 5 positive squares)
%K A215537 nonn
%O A215537 1,1
%A A215537 _Jon Perry_, Aug 15 2012