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, (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))
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
STATUS
approved