A039955 Squarefree numbers congruent to 1 (mod 4).

1, 5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 129, 133, 137, 141, 145, 149, 157, 161, 165, 173, 177, 181, 185, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 249, 253, 257, 265, 269

Offset

1,2

Comments

The subsequence of primes is A002144.

The subsequence of semiprimes (intersection with A001358) begins: 21, 33, 57, 65, 69, 77, 85, 93, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265.

The subsequence with more than two prime factors (intersection with A033942) begins: 105 = 3 * 5 * 7, 165 = 3 * 5 * 11, 273, 285, 345, 357, 385, 429, 465. - Jonathan Vos Post, Feb 19 2011

Except for a(1) = 1 these are the squarefree members of A079896 (i.e., squarefree determinants D of indefinite binary quadratic forms). - Wolfdieter Lang, Jun 01 2013

The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

References

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A. M. Legendre, Diviseurs de la formule t^2+a*u^2, a étant de la forme 4 n + 1, Essai sur la Théorie des Nombres An VI, Table IV. See first column. [Paul Curtz, Aug 14 2019]

Mathematica

fQ[n_] := Max[Last /@ FactorInteger@ n] == 1 && Mod[n, 4] == 1; Select[ Range@ 272, fQ] (* Robert G. Wilson v *)
Select[Range[1, 300, 4], SquareFreeQ[#]&] (* Harvey P. Dale, Mar 27 2020 *)

Prog

(Magma) [4*n+1: n in [0..67] | IsSquarefree(4*n+1)];  // Bruno Berselli, Mar 03 2011
(Haskell)
a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011
(PARI) list(lim)=my(v=List([1])); forfactored(n=5, lim\1, if(vecmax(n[2][, 2])==1 && n[1]%4==1, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017
(PARI) is(n)=n%4==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

Crossrefs

Cf. A002144, A039956, A039957, A005117, A185197.

Sequence in context: A226165 A166409 A077425 * A213340 A014539 A249034

Adjacent sequences: A039952 A039953 A039954 * A039956 A039957 A039958

Keyword

nonn,easy,nice

Author

R. K. Guy

Status

approved