OFFSET
1,2
COMMENTS
Primitively but not imprimitively represented by x^2 + y^2.
Squarefree members of A202057. - Artur Jasinski, Dec 10 2011
Union of A231754 and 2*A231754. Squarefree numbers whose prime factors are in A002313. - Robert Israel, Aug 23 2017
It appears that a(n) is the n-th index, k, such that f(k) = 2, where f(k) = 3*(Sum_{i=1..k} floor(i^2/k)) - k^2 (see A175908). - John W. Layman, May 16 2011
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see page 123.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
FORMULA
a(n) ~ k*n*sqrt(log n), where k = 2.1524249... = A013661/A064533. - Charles R Greathouse IV, Apr 20 2015
MAPLE
N:= 1000: # to get all terms <= N
R:= {1, 2}:
p:= 2:
do
p:= nextprime(p);
if p > N then break fi;
if p mod 4 <> 1 then next fi;
R:= R union select(`<=`, map(`*`, R, p), N);
od:
sort(convert(R, list)); # Robert Israel, Aug 23 2017
MATHEMATICA
lim = 17; t = Join[{1}, Select[Union[Flatten[Table[x^2 + y^2, {x, lim}, {y, x}]]], # < lim^2 && SquareFreeQ[#] &]]
Select[Union[Total/@Tuples[Range[0, 20]^2, 2]], SquareFreeQ] (* Harvey P. Dale, Jul 26 2017 *)
Block[{nn = 350, p}, p = {1, 2}~Join~Select[Prime@ Range@ PrimePi@ nn, Mod[#, 4] == 1 &]; Select[Range@ nn, And[SquareFreeQ@ #, SubsetQ[p, FactorInteger[#][[All, 1]]]] &]] (* Michael De Vlieger, Aug 23 2017 *)
(* or *)
Select[Range[350], SquareFreeQ@ # && ! MemberQ[Mod[First /@ FactorInteger@ #, 4], 3] &] (* Giovanni Resta, Aug 25 2017 *)
PROG
(PARI) is(n)=my(f=factor(n)); for(i=1, #f~, if(f[i, 2]>1 || f[i, 1]%4==3, return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015
(Haskell)
a020893 n = a020893_list !! (n-1)
a020893_list = filter (\x -> any (== 1) $ map (a010052 . (x -)) $
takeWhile (<= x) a000290_list) a005117_list
-- Reinhard Zumkeller, May 28 2015
(Python)
from itertools import count, islice
from sympy import factorint
def A020893_gen(): # generator of terms
return filter(lambda n:all(p & 3 != 3 and e == 1 for p, e in factorint(n).items()), count(1))
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Aug 30 2017
STATUS
approved