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A057948
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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.
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10
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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
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OFFSET
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1,1
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COMMENTS
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Factorization in S is not unique. See related sequences.
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
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
A. I. Kostrikin, Introduction to Algebra, universitext, Springer, 1982.
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LINKS
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EXAMPLE
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21 is of the form 4i+1, but it is not divisible by any smaller S-primes, so 21 is in the sequence.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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