cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

Original entry on oeis.org

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
Offset: 1

Views

Author

Steven Finch

Keywords

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  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see page 123.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

Crossrefs

Cf. A001481, A008784, A022544, A034023, A175908.
Cf. also A000290, A010052, A005117, A084349, A002313, A231754.

Programs

  • 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
  • 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 *)
  • 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
  • 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))
A020893_list = list(islice(A020893_gen(),30)) # Chai Wah Wu, Jun 28 2022

Formula

a(n) ~ k*n*sqrt(log n), where k = 2.1524249... = A013661/A064533. - Charles R Greathouse IV, Apr 20 2015

Extensions

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

A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

Original entry on oeis.org

1, 3, 7, 11, 19, 21, 23, 31, 33, 43, 47, 57, 59, 67, 69, 71, 77, 79, 83, 93, 103, 107, 127, 129, 131, 133, 139, 141, 151, 161, 163, 167, 177, 179, 191, 199, 201, 209, 211, 213, 217, 223, 227, 231, 237, 239, 249, 251, 253, 263, 271, 283, 301, 307, 309, 311, 321, 329
Offset: 1

Views

Author

Arnaud Vernier, Oct 29 2009

Keywords

Comments

Or, numbers that are not divisible by the sum of two squares (other than 1). - Clarified by Gabriel Conant, Apr 18 2016
If a term divides the sum of two squares, then it divides each of the two numbers individually. Moreover, only the numbers in this sequence have this property. See link for proof. - V Sai Prabhav, Jul 15 2025

Links

Crossrefs

Cf. A002145, A004613, A004614, A005117, A231754.
Cf. A175647, A243379.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
S:= {1};
for p from 3 by 4 to N do
  if isprime(p) then
    S:= S union select(`<=`, map(t -> t*p, S),N)
  fi
od:
sort(convert(S,list)); # Robert Israel, Apr 18 2016
  • Mathematica
    Select[Range@ 1000, #==1 || ({{3}, {1}} == Union /@ {Mod[ #[[1]], 4], #[[2]]} &@ Transpose@ FactorInteger@ #) &] (* Giovanni Resta, Apr 18 2016 *)
  • PARI
    isok(n) = if (! issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 3, return (0))); 1 \\ Michel Marcus, Sep 04 2013

Formula

A005117 INTERSECT A004614. - R. J. Mathar, Nov 05 2009
The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A243379/(2*sqrt(A175647)) = 0.4165140462... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

Extensions

Edited by Zak Seidov, Oct 30 2009
Narrowed definition down to squarefree numbers - R. J. Mathar, Nov 05 2009

A231791 Integers k such that A231589(k) = floor(k*(k-1)/4) - k.

Original entry on oeis.org

8, 25, 77, 125, 133, 209, 301, 325, 425, 469, 473, 725, 737, 817, 925, 1025, 1141, 1273, 1325, 1525, 1625, 1793, 1825, 2125, 2225, 2425, 2525, 2725, 2825, 2881, 3097, 3425, 3625, 3725, 3925, 4325, 4525, 4625, 4825, 4925, 5125, 5525, 5725, 5825, 6025, 6425
Offset: 1

Views

Author

Michel Marcus, Nov 13 2013

Keywords

Comments

It appears that this sequence is the union of 3 sets.
First term is 8, and is the only even known value.
Then we get terms that are equal to 25 * b with b a squarefree product of primes congruent to 1 modulo 4 (A002144), that is, terms of A231754.
And we get the following terms 77, 133, 209, 301, 469, 473, 737, 817, 1141, 1273, 1793, 2881, 3097, 7009, 10921. These numbers are the products of 2 distinct primes from this list: 7, 11, 19, 43, 67, 163 (a subsequence of A003173).

Links

Crossrefs

Cf. A231589.

Programs

  • PARI
    isok(n) = A231589(n) == n*(n-1)/4 - n;
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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