A057948 S-primes: let S = {1,5,9, ... 4i+1, ...}; then an S-prime is in S but is not divisible by any members of S except itself and 1.

5, 9, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 121, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, 201, 209, 213, 217, 229, 233, 237, 241, 249, 253, 257, 269, 277, 281, 293, 301, 309, 313, 317, 321, 329

Offset: 1

Author

Jud McCranie, Oct 14 2000

Keywords

• nonn

Comments

Factorization in S is not unique. See related sequences.

Kostrikin calls these numbers quasi-primes. - Arkadiusz Wesolowski, Aug 19 2017

a(n) is a prime of the form 4*n + 1 or a product of 2 primes of the form 4*n + 3. - David A. Corneth, Nov 10 2018

Examples

			21 is of the form 4i+1, but it is not divisible by any smaller S-primes, so 21 is in the sequence.
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
• A. I. Kostrikin, Introduction to Algebra, universitext, Springer, 1982.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Hilbert Number [From _Eric W. Weisstein_, Sep 15 2008]

Crossrefs

Union of A002144 and A107978. - Charlie Neder, Nov 03 2018

Cf. A054520, A057949, A057950.

Programs

Maple

N:= 1000: # to get all terms <= N
S:= {seq(4*i+1,i=1..floor((N-1)/4))}:
for n from 1 while n <= nops(S) do
  r:= S[n];
  S:= S minus {seq(i*r,i=2..floor(N/r))};
od:
S; # Robert Israel, Dec 14 2014

Mathematica

nn = 100; Complement[Table[4 k + 1, {k, 1, nn}], Union[Flatten[ Table[Table[(4 k + 1) (4 j + 1), {k, 1, j}], {j, 1, nn}]]]] (* Geoffrey Critzer, Dec 14 2014 *)

PARI

is(n) = if(n % 2 == 0, return(0)); if(n%4 == 1 && isprime(n), return(1)); f = factor(n); if(vecsum(f[, 2]) != 2, return(0)); for(i = 1, #f[, 1], if(f[i, 1] % 4 == 1, return(0))); n>1 \\ David A. Corneth, Nov 10 2018

Formula

a(n) ~ C n log n / log log n, where C > 2. - Thomas Ordowski, Sep 09 2012

Extensions

Offset corrected by Charlie Neder, Nov 03 2018