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A002479
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Numbers of the form x^2 + 2*y^2.
(Formerly M0547 N0197)
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49
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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
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OFFSET
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1,3
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COMMENTS
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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
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
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REFERENCES
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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).
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LINKS
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MAPLE
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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:
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MATHEMATICA
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Union[#[[1]]+2#[[2]]&/@Tuples[Range[0, 10]^2, 2]] (* Harvey P. Dale, Nov 24 2014 *)
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PROG
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(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)
(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))
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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STATUS
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approved
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