

A022544


Numbers that are not the sum of 2 squares.


59



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
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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
Integers with an equal number of 4k+1 and 4k+3 divisors.  Ant King, Oct 05 2010
There are arbitrarily long runs of consecutive terms. Record runs start at 3, 6, 21, 75, ... (A260157).  Ivan Neretin, Nov 09 2015
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. 98104.


LINKS



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^2n, 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 !! (n1)
a022544_list = elemIndices 0 a000161_list
(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)
(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))


CROSSREFS



KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



