%I #89 Oct 03 2023 10:28:20

%S 3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,46,

%T 47,48,51,54,55,56,57,59,60,62,63,66,67,69,70,71,75,76,77,78,79,83,84,

%U 86,87,88,91,92,93,94,95,96,99,102,103,105,107,108,110,111,112,114,115,118,119,120,123,124,126,127,129,131,132,133,134,135,138,139,140,141,142,143,147,150,151,152,154,155,156,158,159,161,163,165,166,167,168,171,172,174,175,176,177,179,182,183,184,186,187,188,189,190,191,192,195,198,199

%N Numbers that are not the sum of 2 squares.

%C Conjecture: if k is not the sum of 2 squares then sigma(k) == 0 (mod 4) (the converse does not hold, as demonstrated by the entries in A025303). - _Benoit Cloitre_, May 19 2002

%C Numbers having some prime factor p == 3 (mod 4) to an odd power. sigma(n) == 0 (mod 4) because of this prime factor. Every k == 3 (mod 4) is a term. First differences are always 1, 2, 3 or 4, each occurring infinitely often. - _David W. Wilson_, Mar 09 2005

%C Complement of A000415 in the nonsquare positive integers A000037. - _Max Alekseyev_, Jan 21 2010

%C Integers with an equal number of 4k+1 and 4k+3 divisors. - _Ant King_, Oct 05 2010

%C A000161(a(n)) = 0; A070176(a(n)) > 0; A046712 is a subsequence. - _Reinhard Zumkeller_, Feb 04 2012, Aug 16 2011

%C There are arbitrarily long runs of consecutive terms. Record runs start at 3, 6, 21, 75, ... (A260157). - _Ivan Neretin_, Nov 09 2015

%C From _Klaus Purath_, Sep 04 2023: (Start)

%C There are no squares in this sequence.

%C There are also no numbers of the form n^2 + 1 (A002522) or n^2 + 4 (A087475).

%C Every term a(n) raised to an odd power belongs to the sequence just as every product of an odd number of terms. This is also true for all integer sequences represented by the indefinite binary quadratic forms a(n)*x^2 - y^2. These sequences also do not contain squares. (End)

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.

%H T. D. Noe, <a href="/A022544/b022544.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/lr/lr.html">Landau-Ramanujan Constant</a> [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010605004309/http://www.mathsoft.com/asolve/constant/lr/lr.html">Landau-Ramanujan Constant</a> [From the Wayback machine]

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F Limit_{n->oo} a(n)/n = 1.

%t Select[Range[199], Length[PowersRepresentations[ #, 2, 2]] == 0 &] (* _Ant King_, Oct 05 2010 *)

%t Select[Range[200],SquaresR[2,#]==0&] (* _Harvey P. Dale_, Apr 21 2012 *)

%o (PARI) for(n=0,200,if(sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1)))==0,print1((n),",")))

%o (PARI) is(n)=if(n%4==3, return(1)); my(f=factor(n)); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0 \\ _Charles R Greathouse IV_, Sep 01 2015

%o (Haskell)

%o import Data.List (elemIndices)

%o a022544 n = a022544_list !! (n-1)

%o a022544_list = elemIndices 0 a000161_list

%o -- _Reinhard Zumkeller_, Aug 16 2011

%o (Magma) [n: n in [0..160] | NormEquation(1, n) eq false]; // _Vincenzo Librandi_, Jan 15 2017

%o (Python)

%o def aupto(lim):

%o squares = [k*k for k in range(int(lim**.5)+2) if k*k <= lim]

%o sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])

%o return sorted(set(range(lim+1)) - sum2sqs)

%o print(aupto(199)) # _Michael S. Branicky_, Mar 06 2021

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A022544_gen(): # generator of terms

%o return filter(lambda n:any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()),count(0))

%o A022544_list = list(islice(A022544_gen(),30)) # _Chai Wah Wu_, Jun 28 2022

%Y Complement of A001481; subsequence of A111909.

%Y Cf. A018825, A025284, A000404, A007692.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Benoit Cloitre_, May 19 2002