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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Factorization in S is not unique. See related sequences.

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

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

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

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]

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

Cf. A054520, A057949, A057950.

Sequence in context: A314680 A314681 A291180 * A314682 A004958 A190887

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

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Last modified May 21 18:43 EDT 2019. Contains 323444 sequences. (Running on oeis4.)