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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.

%I A054520 #33 Jul 02 2025 16:01:59
%S A054520 1,25,45,65,81,85,105,117,125,145,153,165,169,185,189,205,221,225,245,
%T A054520 261,265,273,285,289,297,305,325,333,345,357,365,369,377,385,405,425,
%U A054520 429,441,445,465,477,481,485,493,505,513,525,533,545,549,561,565,585
%N 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.
%C A054520 The set S is a standard example of a set where unique factorization does not hold.
%C A054520 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
%D A054520 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
%H A054520 William A. Tedeschi, <a href="/A054520/b054520.txt">Table of n, a(n) for n = 1..10000</a>
%H A054520 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertNumber.html">Hilbert Number</a>
%e A054520 49 is an S-prime.
%t A054520 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 *)
%o A054520 (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
%Y A054520 Cf. A057948, A057949, A057950.
%K A054520 nonn,nice,easy
%O A054520 1,2
%A A054520 _N. J. A. Sloane_, Apr 09 2000
%E A054520 More terms from _James Sellers_, Apr 11 2000
%E A054520 Offset corrected by _Andrew Howroyd_, Nov 25 2018