

A091236


Nonprimes of form 4k+3.


12



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
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OFFSET

1,1


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. 1516 for the proof of Jacobi.  Wolfdieter Lang, Jul 29 2016


REFERENCES

E. Grosswald, Representations of Integers as Sums of Squares. SpringerVerlag, NY, 1985.


LINKS

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


EXAMPLE

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


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 *)


PROG

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


CROSSREFS

Cf. A002145, A004767, A078703.
Cf. A091113A092256.
Sequence in context: A080945 A080946 A253056 * A307854 A053177 A145195
Adjacent sequences: A091233 A091234 A091235 * A091237 A091238 A091239


KEYWORD

easy,nonn


AUTHOR

Labos Elemer, Feb 24 2004


STATUS

approved



