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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Crossrefs

Complement of A001481; subsequence of A111909.

Cf. A018825, A025284, A000404, A007692.

Programs

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

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 *)

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

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

Formula

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

Extensions

More terms from Benoit Cloitre, May 19 2002