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A022544 Numbers that are not the sum of 2 squares.

%I A022544 #89 Oct 03 2023 10:28:20
%S A022544 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 A022544 47,48,51,54,55,56,57,59,60,62,63,66,67,69,70,71,75,76,77,78,79,83,84,
%U A022544 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 A022544 Numbers that are not the sum of 2 squares.
%C A022544 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 A022544 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 A022544 Complement of A000415 in the nonsquare positive integers A000037. - _Max Alekseyev_, Jan 21 2010
%C A022544 Integers with an equal number of 4k+1 and 4k+3 divisors. - _Ant King_, Oct 05 2010
%C A022544 A000161(a(n)) = 0; A070176(a(n)) > 0; A046712 is a subsequence. - _Reinhard Zumkeller_, Feb 04 2012, Aug 16 2011
%C A022544 There are arbitrarily long runs of consecutive terms. Record runs start at 3, 6, 21, 75, ... (A260157). - _Ivan Neretin_, Nov 09 2015
%C A022544 From _Klaus Purath_, Sep 04 2023: (Start)
%C A022544 There are no squares in this sequence.
%C A022544 There are also no numbers of the form n^2 + 1 (A002522) or n^2 + 4 (A087475).
%C A022544 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 A022544 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
%H A022544 T. D. Noe, <a href="/A022544/b022544.txt">Table of n, a(n) for n = 1..10000</a>
%H A022544 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/lr/lr.html">Landau-Ramanujan Constant</a> [Broken link]
%H A022544 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 A022544 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A022544 Limit_{n->oo} a(n)/n = 1.
%t A022544 Select[Range[199], Length[PowersRepresentations[ #, 2, 2]] == 0 &] (* _Ant King_, Oct 05 2010 *)
%t A022544 Select[Range[200],SquaresR[2,#]==0&] (* _Harvey P. Dale_, Apr 21 2012 *)
%o A022544 (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 A022544 (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 A022544 (Haskell)
%o A022544 import Data.List (elemIndices)
%o A022544 a022544 n = a022544_list !! (n-1)
%o A022544 a022544_list = elemIndices 0 a000161_list
%o A022544 -- _Reinhard Zumkeller_, Aug 16 2011
%o A022544 (Magma) [n: n in [0..160] | NormEquation(1, n) eq false]; // _Vincenzo Librandi_, Jan 15 2017
%o A022544 (Python)
%o A022544 def aupto(lim):
%o A022544   squares = [k*k for k in range(int(lim**.5)+2) if k*k <= lim]
%o A022544   sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
%o A022544   return sorted(set(range(lim+1)) - sum2sqs)
%o A022544 print(aupto(199)) # _Michael S. Branicky_, Mar 06 2021
%o A022544 (Python)
%o A022544 from itertools import count, islice
%o A022544 from sympy import factorint
%o A022544 def A022544_gen(): # generator of terms
%o A022544     return filter(lambda n:any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()),count(0))
%o A022544 A022544_list = list(islice(A022544_gen(),30)) # _Chai Wah Wu_, Jun 28 2022
%Y A022544 Complement of A001481; subsequence of A111909.
%Y A022544 Cf. A018825, A025284, A000404, A007692.
%K A022544 nonn,nice
%O A022544 1,1
%A A022544 _N. J. A. Sloane_
%E A022544 More terms from _Benoit Cloitre_, May 19 2002