A125853 Squared radii of circles centered at a grid point in a square lattice hitting exactly 4 points. Indices k such that A004018(k)=4.

1, 2, 4, 8, 9, 16, 18, 32, 36, 49, 64, 72, 81, 98, 121, 128, 144, 162, 196, 242, 256, 288, 324, 361, 392, 441, 484, 512, 529, 576, 648, 722, 729, 784, 882, 961, 968, 1024, 1058, 1089, 1152, 1296, 1444, 1458, 1568, 1764, 1849, 1922, 1936, 2048, 2116, 2178, 2209

Offset: 1

Author

Hugo Pfoertner, Jan 07 2007

Keywords

• nonn

Comments

From Jean-Christophe Hervé, Nov 17 2013: (Start)

Squares or double of squares that are not sum of two distinct nonzero squares.

Numbers without prime factor of form 4*k+1 and without prime factor of form 4*k+3 to an odd multiplicity.

The sequence is closed under multiplication. Primitive elements are 1, 2 and square of primes of form 4*k+3, that is union of {1, 2} and A087691.

Sequence A001481 (sum of two squares) is the union of {0}, this sequence and A004431 (sum of two distinct nonzero squares). These 4 sequences are all closed under multiplication. (End)

Links

• Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 18 2007, Table of n, a(n) for n = 1..501
• G. Xiao, Two squares
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A004018, A001481, A004431.

Programs

PARI

for(n=1,100000,fctrs=factor(n);c=1;for(i=1,matsize(fctrs)[1],p4=fctrs[i,1]%4;if(p4==1 || (p4==3 && fctrs[i,2]%2==1), c=0)); if(c,print1(n","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 17 2007

Formula

Numbers of the form 2^e0 * 3^(2*e1) * 7^(2*e2) * 11^(2*e3) * ... * qk^(2*ek) where qk is the k-th prime of the form 4*n+3 (A002145). - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 17 2007