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.

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

Offset

1,2

Comments

The set S is a standard example of a set where unique factorization does not hold.

With the exception a(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 * A343826 A339958 A192261

Adjacent sequences: A054517 A054518 A054519 * A054521 A054522 A054523

Keyword

nonn,nice,easy,changed

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