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A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3. 9
1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 146, 149, 157, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 305, 313, 314, 317, 337, 346, 349 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Primitively but not imprimitively represented by x^2+y^2.

The disjoint union of {1}, A003654, and A031398. - Max Alekseyev, Mar 09 2010

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; See page 123.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]

Index entries for sequences related to sums of squares

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

CROSSREFS

Cf. A001481, A008784, A022544, A034023, A175908.

Cf. also A000290, A010052, A005117, A084349, A002313, A231754.

Sequence in context: A008784 A224450 A226828 * A281292 A145017 A031396

Adjacent sequences:  A020890 A020891 A020892 * A020894 A020895 A020896

KEYWORD

nonn

AUTHOR

Steven Finch

EXTENSIONS

Edited by N. J. A. Sloane, Aug 30 2017

STATUS

approved

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Last modified August 15 11:48 EDT 2020. Contains 336492 sequences. (Running on oeis4.)