A091236 Nonprimes of form 4k+3.

15, 27, 35, 39, 51, 55, 63, 75, 87, 91, 95, 99, 111, 115, 119, 123, 135, 143, 147, 155, 159, 171, 175, 183, 187, 195, 203, 207, 215, 219, 231, 235, 243, 247, 255, 259, 267, 275, 279, 287, 291, 295, 299, 303, 315, 319, 323, 327, 335, 339, 343, 351, 355, 363

Offset: 1

Author

Labos Elemer, Feb 24 2004

Keywords

• easy
• nonn

Comments

If we define f(n) to be the number of primes (counted with multiplicity) of the form 4k + 3 that divide n, then with this sequence f(a(n)) is always odd. For example, 95 is divisible by 17 and 99 is divisible by 3 (twice) and 11. - Alonso del Arte, Jan 13 2016

Complement of A002145 with respect to A004767. - Michel Marcus, Jan 17 2016

With the Jan 05 2004 Jovovic comment on A078703: The number of 1 and -1 (mod 4) divisors of a(n) are identical. Proof: each number 3 (mod 4) is trivially not a sum of two squares. The number of solutions of n as a sum of two squares is r_2(n) = 4*(d_1(n) - d_3(n)), where d_k(n) is the number of k (mod 4) divisors of n. See e.g., Grosswald, pp. 15-16 for the proof of Jacobi. - Wolfdieter Lang, Jul 29 2016

Examples

			27 = 4 * 6 + 3 = 3^3.
35 = 4 * 8 + 3 = 5 * 7.
a(8) = 75 with 2*A078703(19) = 6 divisors [1, 3, 5, 15, 25, 75], which are 1, -1, 1, -1, 1, -1 (mod 4). - _Wolfdieter Lang_, Jul 29 2016
		

References

• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Links

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

Crossrefs

Cf. A002145, A004767, A078703.

Cf. A091113-A092256.

Programs

Maple

A091236 := proc(n)
    option remember  ;
    local a;
    if n = 1 then
        15 ;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and modp(a,4) = 3 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A091236(n),n=1..20) ; # R. J. Mathar, Jul 20 2025

Mathematica

Select[Range[1000], !PrimeQ[#] && IntegerQ[(# - 3)/4] &] (* Harvey P. Dale, Aug 16 2013 *)
Select[4Range[100] - 1, Not[PrimeQ[#]] &] (* Alonso del Arte, Jan 13 2016 *)

PARI

lista(nn) = for(n=1, nn, if(!isprime(k=4*n+3), print1(k, ", "))); \\ Altug Alkan, Jan 17 2016