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

1,1

Comments

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

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 305.

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

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).

Mathematica

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

Prog

(Haskell)
a000419 n = a000419_list !! (n-1)
a000419_list = filter ((== 3) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015
(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)
from math import isqrt
def aupto(lim):
  squares = [k*k for k in range(1, isqrt(lim)+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

Crossrefs

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

Sequence in context: A066157 A073159 A395242 * A353716 A178890 A332933

Adjacent sequences: A000416 A000417 A000418 * A000420 A000421 A000422

Keyword

nonn,nice,easy

Author

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

Extensions

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

Status

approved