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

3, 6, 11, 12, 14, 19, 21, 22, 24, 27, 30, 33, 35, 38, 42, 43, 44, 46, 48, 51, 54, 56, 57, 59, 62, 66, 67, 69, 70, 75, 76, 77, 78, 83, 84, 86, 88, 91, 93, 94, 96, 99, 102, 105, 107, 108, 110, 114, 115, 118, 120, 123, 126, 129, 131, 132, 133, 134, 138, 139, 140, 141, 142

Offset: 1

Author

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

Keywords

• nonn
• nice
• easy

Comments

A002828(a(n)) = 3; A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Square Number.
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000378, A000408, A000415, A002828, A004215, A025427.

Programs

Haskell

a000419 n = a000419_list !! (n-1)
a000419_list = filter ((== 3) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015

Mathematica

Select[Range[150],SquaresR[3,#]>0&&SquaresR[2,#]==0&] (* Harvey P. Dale, Nov 01 2011 *)

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

Python

def aupto(lim):
  squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
  sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
  sum3sqs = set(a+b for a in sum2sqs for b in squares)
  return sorted(set(range(lim+1)) & (sum3sqs - sum2sqs - set(squares)))
print(aupto(142)) # Michael S. Branicky, Mar 06 2021

Formula

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).

Extensions

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