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

Author

R. K. Guy

Keywords

• nonn
• easy
• nice

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]

Crossrefs

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

Programs

Haskell

a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

Magma

[4*n+1: n in [0..67] | IsSquarefree(4*n+1)];  // Bruno Berselli, Mar 03 2011

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 *)

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