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A000419 Numbers that are the sum of 3 but no fewer nonzero squares.

%I A000419 #33 Feb 16 2025 08:32:21
%S A000419 3,6,11,12,14,19,21,22,24,27,30,33,35,38,42,43,44,46,48,51,54,56,57,
%T A000419 59,62,66,67,69,70,75,76,77,78,83,84,86,88,91,93,94,96,99,102,105,107,
%U A000419 108,110,114,115,118,120,123,126,129,131,132,133,134,138,139,140,141,142
%N A000419 Numbers that are the sum of 3 but no fewer nonzero squares.
%C A000419 A002828(a(n)) = 3; A025427(a(n)) > 0. - _Reinhard Zumkeller_, Feb 26 2015
%D A000419 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.
%H A000419 Ray Chandler, <a href="/A000419/b000419.txt">Table of n, a(n) for n = 1..10000</a>
%H A000419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number.</a>
%H A000419 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A000419 Legendre: a nonnegative integer is a sum of three (or fewer) squares iff it is not of the form 4^k m with m == 7 (mod 8).
%t A000419 Select[Range[150],SquaresR[3,#]>0&&SquaresR[2,#]==0&] (* _Harvey P. Dale_, Nov 01 2011 *)
%o A000419 (Haskell)
%o A000419 a000419 n = a000419_list !! (n-1)
%o A000419 a000419_list = filter ((== 3) . a002828) [1..]
%o A000419 -- _Reinhard Zumkeller_, Feb 26 2015
%o A000419 (PARI) is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return( n/4^valuation(n,4)%8 !=7 ))); 0 \\ _Charles R Greathouse IV_, Feb 07 2017
%o A000419 (Python)
%o A000419 def aupto(lim):
%o A000419   squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
%o A000419   sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
%o A000419   sum3sqs = set(a+b for a in sum2sqs for b in squares)
%o A000419   return sorted(set(range(lim+1)) & (sum3sqs - sum2sqs - set(squares)))
%o A000419 print(aupto(142)) # _Michael S. Branicky_, Mar 06 2021
%Y A000419 Cf. A000378, A000408, A000415, A002828, A004215, A025427.
%K A000419 nonn,nice,easy
%O A000419 1,1
%A A000419 _N. J. A. Sloane_ and _J. H. Conway_
%E A000419 More terms from Arlin Anderson (starship1(AT)gmail.com)