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

Author

N. J. A. Sloane, Apr 09 2000

Keywords

• nonn
• nice
• easy

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

Examples

			49 is an S-prime.
		

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

Crossrefs

Cf. A057948, A057949, A057950.

Programs

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 *)

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

Extensions

More terms from James Sellers, Apr 11 2000

Offset corrected by Andrew Howroyd, Nov 25 2018