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

%I A000415 #47 Feb 16 2025 08:32:21
%S A000415 2,5,8,10,13,17,18,20,26,29,32,34,37,40,41,45,50,52,53,58,61,65,68,72,
%T A000415 73,74,80,82,85,89,90,97,98,101,104,106,109,113,116,117,122,125,128,
%U A000415 130,136,137,145,146,148,149,153,157,160,162,164,170,173,178,180,181
%N A000415 Numbers that are the sum of 2 but no fewer nonzero squares.
%C A000415 Only these numbers can occur as discriminants of quintic polynomials with solvable Galois group F20. - _Artur Jasinski_, Oct 25 2007
%C A000415 Complement of A022544 in the nonsquare positive integers A000037. - _Max Alekseyev_, Jan 21 2010
%C A000415 Nonsquare positive integers D such that Pell equation y^2 - D*x^2 = -1 has rational solutions. - _Max Alekseyev_, Mar 09 2010
%C A000415 Nonsquares for which all 4k+3 primes in the integer's canonical form occur with even multiplicity. - _Ant King_, Nov 02 2010
%D A000415 E. Grosswald, Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), p.15. - _Ant King_, Nov 02 2010
%H A000415 T. D. Noe, <a href="/A000415/b000415.txt">Table of n, a(n) for n = 1..10000</a>
%H A000415 R. K. Guy, <a href="https://www.jstor.org/stable/2324367">Every number is expressible as the sum of how many polygonal numbers?</a>, Amer. Math. Monthly 101 (1994), 169-172. - _Ant King_, Nov 02 2010
%H A000415 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A000415 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A000415 { A000404 } minus { A134422 }. - _Artur Jasinski_, Oct 25 2007
%t A000415 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 *)
%t A000415 Select[Range[181],Length[PowersRepresentations[ #,2,2]]>0 && !IntegerQ[Sqrt[ # ]] &] (* _Ant King_, Nov 02 2010 *)
%o A000415 (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
%o A000415 (Python)
%o A000415 from itertools import count, islice
%o A000415 from sympy import factorint
%o A000415 def A000415_gen(startvalue=2): # generator of terms >= startvalue
%o A000415     for n in count(max(startvalue,2)):
%o A000415         f = factorint(n).items()
%o A000415         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):
%o A000415             yield n
%o A000415 A000415_list = list(islice(A000415_gen(),20)) # _Chai Wah Wu_, Aug 01 2023
%Y A000415 Cf. A000404, A000419, A001481, A002828, A009003, A134422.
%K A000415 nonn,nice
%O A000415 1,1
%A A000415 _N. J. A. Sloane_ and _J. H. Conway_
%E A000415 More terms from Arlin Anderson (starship1(AT)gmail.com)