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
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.
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]
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))
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Benoit Cloitre, May 19 2002
STATUS
approved