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 A054520 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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 REFERENCES T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1. LINKS William A. Tedeschi, Table of n, a(n) for n = 1..10000 Eric Weisstein's World of Mathematics, Hilbert Number EXAMPLE 49 is an S-prime. MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A057948, A057949, A057950. Sequence in context: A105507 A015911 A188005 * A192261 A038811 A028505 Adjacent sequences:  A054517 A054518 A054519 * A054521 A054522 A054523 KEYWORD nonn,nice,easy AUTHOR N. J. A. Sloane, Apr 09 2000 EXTENSIONS More terms from James A. Sellers, Apr 11 2000 Offset corrected by Andrew Howroyd, Nov 25 2018 STATUS approved

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Last modified February 20 15:20 EST 2019. Contains 320336 sequences. (Running on oeis4.)