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A000415
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Numbers that are the sum of 2 but no fewer nonzero squares.
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17
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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
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OFFSET
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1,1
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COMMENTS
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Only these numbers can occur as discriminants of quintic polynomials with solvable Galois group F20. - Artur Jasinski, Oct 25 2007
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
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REFERENCES
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E. Grosswald, Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), p.15. - Ant King, Nov 02 2010
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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More terms from Arlin Anderson (starship1(AT)gmail.com)
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STATUS
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approved
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