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A000047 Number of integers <= 2^n of form x^2 - 2y^2.
(Formerly M0701 N0259)
3
1, 2, 3, 5, 8, 15, 26, 48, 87, 161, 299, 563, 1066, 2030, 3885, 7464, 14384, 27779, 53782, 104359, 202838, 394860, 769777, 1502603, 2936519, 5744932, 11249805, 22048769, 43248623, 84894767, 166758141, 327770275, 644627310, 1268491353, 2497412741 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
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
Seth A. Troisi, Table of n, a(n) for n = 0..50 (terms 0..35 from Ray Chandler, 36..37 from Pontus von Brömssen)
D. Borwein, J. M. Borwein, P. B. Borwein, R. Girgensohn, Giuga's Conjecture on Primality, Am. Math. Monthly 103 (1) (1996), 40-50.
D. Shanks and L. P. Schmid, Variations on a theorem of Landau. Part I, Math. Comp., 20 (1966), 551-569.
Seth A. Troisi, C++ and Python programs
EXAMPLE
There are 5 integers <= 2^3 of form x^2 - 2y^2. The five (x,y) pairs (1,0), (2,1), (2,0), (3,1), (4,2) give respectively: 1, 2, 4, 7, 8. So a(3) = 5. - Bernard Schott, Feb 10 2019
MATHEMATICA
cnt=0; n=0; Table[n++; While[{p, e}=Transpose[FactorInteger[n]]; If[Select[p^e, MemberQ[{3, 5}, Mod[ #, 8]] &] == {}, cnt++ ]; n<2^k, n++ ]; cnt, {k, 0, 20}] (* T. D. Noe, Jan 19 2009 *)
PROG
(PARI) A000047(n)={ local(f, c=0); for(m=1, 2^n, for(i=1, #f=factor(m)~, abs(f[1, i]%8-4)==1 || next; f[2, i]%2 & next(2)); c++); c} \\ See comment in A035251: m=3 or 5 mod 8; M. F. Hasler, Jan 19 2009
CROSSREFS
Cf. A035251.
Sequence in context: A006982 A054539 A026702 * A101172 A192677 A307909
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Giovanni Resta and Harry J. Smith, Jan 24 2009
STATUS
approved

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Last modified March 28 07:46 EDT 2024. Contains 371235 sequences. (Running on oeis4.)