A002479 Numbers of the form x^2 + 2*y^2. (Formerly M0547 N0197)

0, 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 19, 22, 24, 25, 27, 32, 33, 34, 36, 38, 41, 43, 44, 48, 49, 50, 51, 54, 57, 59, 64, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 98, 99, 100, 102, 107, 108, 113, 114, 118, 121, 123, 128, 129, 131

Offset

1,3

Comments

A positive number k belongs to this sequence if and only if every prime p == 5, 7 (mod 8) dividing k occurs to an even power. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002

Norms of numbers in Z[sqrt(-2)]. - Alonso del Arte, Sep 23 2014

Euler (E256) shows that these numbers are closed under multiplication, according to the Euler Archive. - Charles R Greathouse IV, Jun 16 2016

In addition to the previous comment: The proof was already given 1100 years before Euler by Brahmagupta's identity (a^2 + m*b^2)*(c^2 + m*d^2) = (a*c - m*b*d)^2 + m*(a*d + b*c)^2. - Klaus Purath, Oct 07 2023

References

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 421.

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 59.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..3148 (first 1000 terms from T. D. Noe)

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

L. Euler, (E256) Specimen de usu observationum in mathesi pura, Novi Commentarii academiae scientiarum Petropolitanae 6 (1761), pp. 185-230.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Maple

lis:={}; M:=50; M2:=M^2;
for x from 0 to M do for y from 0 to M do
if x^2+2*y^2 <= M2 then lis:={op(lis), x^2+2*y^2}; fi; od: od:
sort(convert(lis, list)); # N. J. A. Sloane, Apr 30 2015

Mathematica

q = 16; imax = q^2; Select[Union[Flatten[Table[x^2 + 2y^2, {y, 0, q/Sqrt[2]}, {x, 0, q}]]], # <= imax &] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
Union[#[[1]]+2#[[2]]&/@Tuples[Range[0, 10]^2, 2]] (* Harvey P. Dale, Nov 24 2014 *)

Prog

(PARI) is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 1]%8>4 && f[i, 2]%2, return(0))); 1 \\ Charles R Greathouse IV, Nov 20 2012
(PARI) list(lim)=my(v=List()); for(a=0, sqrtint(lim\=1), for(b=0, sqrtint((lim-a^2)\2), listput(v, a^2+2*b^2))); Set(v) \\ Charles R Greathouse IV, Jun 16 2016
(Haskell)
a002479 n = a002479_list !! (n-1)
a002479_list = 0 : filter f [1..] where
   f x = all (even . snd) $ filter ((`elem` [5, 7]) . (`mod` 8) . fst) $
                            zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Feb 20 2014
(Magma) [n: n in [0..131] | NormEquation(2, n) eq true]; // Arkadiusz Wesolowski, May 11 2016
(Python)
from itertools import count, islice
from sympy import factorint
def A002479_gen(): # generator of terms
    return filter(lambda n:all(p & 7 < 5 or e & 1 == 0 for p, e in factorint(n).items()), count(0))
A002479_list = list(islice(A002479_gen(), 30)) # Chai Wah Wu, Jun 27 2022

Crossrefs

Complement of A097700. For primes see A033203.

Cf. A035251, A027748, A124010, A003628.

Sequence in context: A020900 A333309 A282697 * A010458 A354620 A218980

Adjacent sequences: A002476 A002477 A002478 * A002480 A002481 A002482

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Status

approved