cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A022544 Numbers that are not the sum of 2 squares.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

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

A111925 Numbers of the form a^2 + b^4, with a,b > 0.

Original entry on oeis.org

2, 5, 10, 17, 20, 25, 26, 32, 37, 41, 50, 52, 65, 80, 82, 85, 90, 97, 101, 106, 116, 117, 122, 130, 137, 145, 160, 162, 170, 181, 185, 197, 202, 212, 225, 226, 241, 250, 257, 260, 265, 272, 277, 281, 290, 292, 305, 306, 320, 325, 337, 340, 356, 362, 370, 377
Offset: 1

Views

Author

Stefan Steinerberger, Nov 25 2005

Keywords

Comments

Subsequence of A000404.
Although there are squares, cubes, fifth powers, ... in this sequence, there are no fourth powers. - Altug Alkan, Apr 09 2016
Also, numbers z such that z^5 = x^2 + y^4 for x, y >= 1. - M. F. Hasler, Apr 16 2018
The Friedlander-Iwaniec theorem states that there are infinitely many prime numbers in this sequence. These primes are in A028916. - Bernard Schott, Mar 09 2019

Examples

			25 = 3^2 + 2^4, so 25 is an element of the sequence.

Links

Crossrefs

Cf. A055394, A022549; complement of A111909; subsequence of A000404.
Cf. A028916 (subsequence of primes).

Programs

  • Maple
    isA111925 := proc(n)
    local a,b ;
    for a from 1 do
        if a^4 >= n then
            return false;
        end if;
        b := n-a^4 ;
        if issqr(b) then
            return true;
        end if;
    end do:
end proc:
A111925 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA111925(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 22 2013
  • Mathematica
    With[{nn=60},Take[Union[First[#]^2+Last[#]^4&/@Tuples[Range[nn],2]],nn]] (* Harvey P. Dale, Jul 09 2014 *)
  • PARI
    list(lim)=my(v=List(),t); lim\=1; for(b=1,sqrtnint(lim-1,4), t=b^4; for(a=1,sqrtint(lim-t), listput(v,t+a^2))); Set(v) \\ Charles R Greathouse IV, Jun 07 2016
  • PARI
    is(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0 \\ Charles R Greathouse IV, Jun 07 2016
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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