A022544 Numbers that are not the sum of 2 squares.

3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 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

Offset

1,1

Comments

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

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

Complement of A000415 in the nonsquare positive integers A000037. - Max Alekseyev, Jan 21 2010

Integers with an equal number of 4k+1 and 4k+3 divisors. - Ant King, Oct 05 2010

A000161(a(n)) = 0; A070176(a(n)) > 0; A046712 is a subsequence. - Reinhard Zumkeller, Feb 04 2012, Aug 16 2011

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

From Klaus Purath, Sep 04 2023: (Start)

There are no squares in this sequence.

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

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)

References

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Steven R. Finch, Landau-Ramanujan Constant [Broken link]

Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]

Index entries for sequences related to sums of squares

Formula

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

Mathematica

Select[Range[199], Length[PowersRepresentations[ #, 2, 2]] == 0 &] (* Ant King, Oct 05 2010 *)
Select[Range[200], SquaresR[2, #]==0&] (* Harvey P. Dale, Apr 21 2012 *)

Prog

(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), ", ")))
(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
(Haskell)
import Data.List (elemIndices)
a022544 n = a022544_list !! (n-1)
a022544_list = elemIndices 0 a000161_list
-- Reinhard Zumkeller, Aug 16 2011
(Magma) [n: n in [0..160] | NormEquation(1, n) eq false]; // Vincenzo Librandi, Jan 15 2017
(Python)
def aupto(lim):
  squares = [k*k for k in range(int(lim**.5)+2) if k*k <= lim]
  sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
  return sorted(set(range(lim+1)) - sum2sqs)
print(aupto(199)) # Michael S. Branicky, Mar 06 2021
(Python)
from itertools import count, islice
from sympy import factorint
def A022544_gen(): # generator of terms
    return filter(lambda n:any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()), count(0))
A022544_list = list(islice(A022544_gen(), 30)) # Chai Wah Wu, Jun 28 2022

Crossrefs

Complement of A001481; subsequence of A111909.

Cf. A018825, A025284, A000404, A007692.

Sequence in context: A101184 A087643 A248150 * A194366 A091067 A269177

Adjacent sequences: A022541 A022542 A022543 * A022545 A022546 A022547

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Benoit Cloitre, May 19 2002

Status

approved