A226165 Squarefree part of A077425(n) (numbers 4*k+1, k>=0, not a square).

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

Offset: 1

Author

Wolfdieter Lang, Jun 14 2013

Keywords

• nonn
• easy

Comments

a(n) == 1 (mod 4), n >= 1. This is because 4*k+1, k>=0, not a square, can only have an even number of odd primes of the type 3 (mod 4) with odd exponents in the prime number factorization. The squarefree part of 4*k+1 has then an even number (maybe 0) of primes of the type 3 (mod 4). Examples:

a(4) = 21 = 3*7, a(6) = 33 = 3*11.

D(n) = A077425(n) are the 1 (mod 4) discriminants of indefinite binary quadratic forms (they are the odd numbers from A079896). sqrt(D(n)) becomes then, up to an integer factor, sqrt(a(n)), which defines a real quadratic number field Q(sqrt(a(n))) with a basis <1, omega(a(n))> for the ring of integers in this field, where omega(a(n)) = (1 + sqrt(a(n)))/2. Example: sqrt(D(9)) = sqrt(45) = 3*sqrt(a(9)) = 3*sqrt(5), with omega(5) = (1 + sqrt(5))/2 (the golden section) for Q(sqrt(5)) = Q(omega(5)).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007913, A077425, A079896.

Programs

Mathematica

SquareFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]); SquareFreePart /@ (4*Range[65] + 1) // DeleteCases[#, 1] & (* Jean-François Alcover, Jun 14 2013 *)

PARI

[core(n) | n <- vector(100,n,4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014

Formula

a(n) = A007913(A077425(n)).