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A002479 Numbers of form x^2 + 2y^2.
(Formerly M0547 N0197)
33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A positive number n belongs to this sequence if and only if every prime p == 5, 7 mod 8 dividing n 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

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

T. D. Noe and 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

CROSSREFS

Complement of A097700. For primes see A033203.

Cf. A035251, A027748, A124010, A003628.

Sequence in context: A186541 A207432 A020900 * A010458 A218980 A084829

Adjacent sequences:  A002476 A002477 A002478 * A002480 A002481 A002482

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 5 15:57 EST 2016. Contains 278770 sequences.