A024508 Numbers that are a sum of 2 distinct nonzero squares in more than one way.

65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 325, 340, 365, 370, 377, 410, 425, 442, 445, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 580, 585, 610, 625, 629, 650, 680, 685, 689, 697, 725, 730, 740, 745, 754, 765, 785, 793, 820, 845, 850, 865, 884, 890, 901, 905, 925, 949, 962, 965, 970, 985, 986, 1000, 1010, 1025, 1037, 1040, 1060, 1066, 1073, 1090, 1105, 1125

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Appears to be n such that sigma(n)==0 (mod 4) and n is expressible as a sum of 2 squares. - Benoit Cloitre, Apr 20 2003

The comment that is in above is true most of the time. However if number of odd divisors of n that is a term of this sequence is not divisible by 4, then sigma(n) cannot be divisible by 4. For example; 325, 425, 625, 650, ... See also A000443 for more related examples. - Altug Alkan, Jun 09 2016

If m is a term then (a^2 + b^2) * m is a term for a,b > 0. Hence this sequence is closed under multiplication. - David A. Corneth, Jun 10 2016

Links

• David A. Corneth, Table of n, a(n) for n = 1..10749
• G. Xiao, Two squares
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A001481, A025303 (exactly 2 ways), A025304 (exactly 3 ways), A025305 (exactly 4 ways), A025306 (exactly 5 ways).

Programs

Mathematica

lst={};q=-1;k=1;Do[Do[x=a^2;Do[y=b^2;If[x+y==n,If[n==q&&k==1,AppendTo[lst,n]];If[n!=q,q=n;k=1,k++ ]],{b,Floor[(n-x)^(1/2)],a+1,-1}],{a,Floor[n^(1/2)],1,-1}],{n,2*6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)

PARI

is(n) = {my(t=0,i);t=sum(i=1,sqrtint((n-1)\2),issquare(n-i^2));t>1} \\ David A. Corneth, Jun 10 2016

PARI

is(n)=if(n<9,return(0)); my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2-issquare(n/2)>1 \\ Charles R Greathouse IV, Jun 10 2016