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A091236
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Nonprimes of form 4k+3.
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12
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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
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1,1
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COMMENTS
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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
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
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) lista(nn) = for(n=1, nn, if(!isprime(k=4*n+3), print1(k, ", "))); \\ Altug Alkan, Jan 17 2016
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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