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A054520
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Let S = {1,5,9,13,..., 4n+1, ...} and call p in S an S-prime if p>1 and the only divisors of p in S are 1 and p; sequence gives elements of S that are not S-primes.
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9
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1, 25, 45, 65, 81, 85, 105, 117, 125, 145, 153, 165, 169, 185, 189, 205, 221, 225, 245, 261, 265, 273, 285, 289, 297, 305, 325, 333, 345, 357, 365, 369, 377, 385, 405, 425, 429, 441, 445, 465, 477, 481, 485, 493, 505, 513, 525, 533, 545, 549, 561, 565, 585
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OFFSET
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1,2
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COMMENTS
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The set S is a standard example of a set where unique factorization does not hold.
With the exception A054520(1)=1, numbers of the form 4*(m + n + 4 m n)+1 (m,n>0). No such number can be prime because 4*(m + n + 4 m n)+1=(4m+1)(4n+1). - Artur Jasinski, Sep 22 2008
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
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LINKS
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EXAMPLE
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49 is an S-prime.
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MATHEMATICA
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a = {}; Do[Do[AppendTo[a, 4(m + n + 4 m n)+1], {m, 1, 100}], {n, 1, 100}]; Union[a] (* Artur Jasinski, Sep 22 2008 *)
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PROG
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(PARI) ok(n)={if(n%4==1, my(f=factor(n)); 2<>sum(i=1, #f~, f[i, 2]*if(f[i, 1]%4==3, 1, 2)), 0)} \\ Andrew Howroyd, Nov 25 2018
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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