cp's OEIS Frontend

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.

%I A125853 #17 Aug 15 2025 00:58:52
%S A125853 1,2,4,8,9,16,18,32,36,49,64,72,81,98,121,128,144,162,196,242,256,288,
%T A125853 324,361,392,441,484,512,529,576,648,722,729,784,882,961,968,1024,
%U A125853 1058,1089,1152,1296,1444,1458,1568,1764,1849,1922,1936,2048,2116,2178,2209
%N 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.
%C A125853 From _Jean-Christophe Hervé_, Nov 17 2013: (Start)
%C A125853 Squares or double of squares that are not sum of two distinct nonzero squares.
%C A125853 Numbers without prime factor of form 4*k+1 and without prime factor of form 4*k+3 to an odd multiplicity.
%C A125853 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.
%C A125853 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)
%H A125853 Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 18 2007, <a href="/A125853/b125853.txt">Table of n, a(n) for n = 1..501</a>
%H A125853 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H A125853 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A125853 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
%o A125853 (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
%Y A125853 Cf. A004018, A001481, A004431.
%K A125853 nonn
%O A125853 1,2
%A A125853 _Hugo Pfoertner_, Jan 07 2007