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

1,1

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

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]

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A054520, A057949, A057950.

Sequence in context: A314680 A314681 A291180 * A314682 A379040 A004958

Adjacent sequences: A057945 A057946 A057947 * A057949 A057950 A057951

Keyword

nonn

Author

Jud McCranie, Oct 14 2000

Extensions

Offset corrected by Charlie Neder, Nov 03 2018

Status

approved