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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved