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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.

%I A057948 #52 Aug 19 2025 16:36:55
%S A057948 5,9,13,17,21,29,33,37,41,49,53,57,61,69,73,77,89,93,97,101,109,113,
%T A057948 121,129,133,137,141,149,157,161,173,177,181,193,197,201,209,213,217,
%U A057948 229,233,237,241,249,253,257,269,277,281,293,301,309,313,317,321,329
%N 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.
%C A057948 Factorization in S is not unique. See related sequences.
%C A057948 Kostrikin calls these numbers quasi-primes. - _Arkadiusz Wesolowski_, Aug 19 2017
%C A057948 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
%D A057948 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
%D A057948 A. I. Kostrikin, Introduction to Algebra, universitext, Springer, 1982.
%H A057948 Robert Israel, <a href="/A057948/b057948.txt">Table of n, a(n) for n = 1..10000</a>
%H A057948 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertNumber.html">Hilbert Number</a> [From _Eric W. Weisstein_, Sep 15 2008]
%F A057948 a(n) ~ C n log n / log log n, where C > 2. - _Thomas Ordowski_, Sep 09 2012
%e A057948 21 is of the form 4i+1, but it is not divisible by any smaller S-primes, so 21 is in the sequence.
%p A057948 N:= 1000: # to get all terms <= N
%p A057948 S:= {seq(4*i+1,i=1..floor((N-1)/4))}:
%p A057948 for n from 1 while n <= nops(S) do
%p A057948   r:= S[n];
%p A057948   S:= S minus {seq(i*r,i=2..floor(N/r))};
%p A057948 od:
%p A057948 S; # _Robert Israel_, Dec 14 2014
%t A057948 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 *)
%o A057948 (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
%Y A057948 Union of A002144 and A107978. - _Charlie Neder_, Nov 03 2018
%Y A057948 Cf. A054520, A057949, A057950.
%K A057948 nonn
%O A057948 1,1
%A A057948 _Jud McCranie_, Oct 14 2000
%E A057948 Offset corrected by _Charlie Neder_, Nov 03 2018