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A359702
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Odd primes p that are not congruent to 2*k modulo prime(k+1) for any positive integer k.
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0
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3, 7, 31, 37, 43, 61, 67, 73, 157, 211, 271, 277, 331, 367, 421, 457, 571, 691, 823, 883, 997, 1093, 1201, 1237, 1303, 1657, 1783, 2053, 2287, 2347, 2371, 2377, 2557, 2803, 2971, 3001, 3061, 3067, 3307, 3313, 3391, 3967, 4021, 4231, 4273, 4357, 4447, 4561, 4603
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OFFSET
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1,1
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COMMENTS
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This sequence arises from a more general study. First, consider a function f : P -> N (where P is the set of the odd prime numbers) such that 0 <= f(p) < p. Then, remove from the set P each prime number q such that q = f(p) (mod p) for some p.
For example, if f(p) = 0 for each p, then the final set is the empty set.
If f(p) = 1 for each p, then the final set seems to be the set of Fermat primes (empirical observation).
If f(p) = p-1, then the final set seems to be the set of Mersenne primes (empirical observation).
For the particular choice f(p) = 2k (where p is the k-th odd prime) this sequence is obtained.
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LINKS
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EXAMPLE
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Terms in this sequence are those odd primes that are neither congruent to 2 (mod 3), nor congruent to 4 (mod 5), nor congruent to 6 (mod 7), nor congruent to 8 (mod 11), etc.
7 is a term because 7 == 1 (mod 3) and 7 == 2 (mod 5).
11 is not a term because 11 == 2 (mod 3).
13 is not a term because 13 == 6 (mod 7).
17 is not a term because 17 == 2 (mod 3).
19 is not a term because 19 == 8 (mod 11).
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PROG
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(PARI) isok(p) = {if(!isprime(p)||p==2, 0, my(k=0); forprime(q=3, p-1, k+=2; if(p%q==k, return(0))); 1) } \\ Andrew Howroyd, Jan 11 2023
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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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