A000415 Numbers that are the sum of 2 but no fewer nonzero squares.

2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 170, 173, 178, 180, 181

Offset: 1

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• nice

Comments

Only these numbers can occur as discriminants of quintic polynomials with solvable Galois group F20. - Artur Jasinski, Oct 25 2007

Complement of A022544 in the nonsquare positive integers A000037. - Max Alekseyev, Jan 21 2010

Nonsquare positive integers D such that Pell equation y^2 - D*x^2 = -1 has rational solutions. - Max Alekseyev, Mar 09 2010

Nonsquares for which all 4k+3 primes in the integer's canonical form occur with even multiplicity. - Ant King, Nov 02 2010

References

• E. Grosswald, Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), p.15. - Ant King, Nov 02 2010

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172. - _Ant King_, Nov 02 2010
• Eric Weisstein's World of Mathematics, Square Number
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000404, A000419, A001481, A002828, A009003, A134422.

Programs

Mathematica

c = {}; Do[Do[k = a^2 + b^2; If[IntegerQ[Sqrt[k]], Null, AppendTo[c,k]], {a, 1, 100}], {b, 1, 100}]; Union[c] (* Artur Jasinski, Oct 25 2007 *)
Select[Range[181],Length[PowersRepresentations[ #,2,2]]>0 && !IntegerQ[Sqrt[ # ]] &] (* Ant King, Nov 02 2010 *)

PARI

is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); !issquare(n) \\ Charles R Greathouse IV, Feb 07 2017

Python

from itertools import count, islice
from sympy import factorint
def A000415_gen(startvalue=2): # generator of terms >= startvalue
    for n in count(max(startvalue,2)):
        f = factorint(n).items()
        if any(e&1 for p,e in f if p&3<3) and not any(e&1 for p,e in f if p&3==3):
            yield n
A000415_list = list(islice(A000415_gen(),20)) # Chai Wah Wu, Aug 01 2023

Formula

{ A000404 } minus { A134422 }. - Artur Jasinski, Oct 25 2007

Extensions

More terms from Arlin Anderson (starship1(AT)gmail.com)