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A106639
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Distinguished primes.
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3
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2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 43, 59, 61, 67, 83, 157, 173, 227, 277, 283, 317, 347, 563, 653, 733, 787, 877, 907, 997, 1213, 1237, 1283, 1307, 1523, 1867, 2083, 2693, 2797, 2803, 3253, 3413, 3517, 3643, 3677, 3733, 3803, 4253, 4363, 4547, 4723, 5387
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OFFSET
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1,1
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COMMENTS
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Primes are distinguished among the integers by having the fewest possible divisors. Among the primes, which primes are similarly distinguished? The distinguished primes have the fewest possible divisors in the neighborhood. Specifically, p is a distinguished prime iff together p-1, p and p+1, have 7 or fewer prime factors, counting multiple factors. Of course, the definition could be adjusted to make 3, or even 2, the unique distinguished prime, but then the sequence of distinguished primes would be severely truncated.
a(1)-a(6) are the only members with fewer than 7 prime factors between p-1, p, and p+1. Dickson's conjecture implies that this sequence is infinite. The Bateman-Horn-Stemmler conjecture suggests that there are about 1.905x/(log x)^3 members up to x. - Charles R Greathouse IV, Apr 20 2011
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LINKS
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FORMULA
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Primes p such that Omega(p^3 - p) <= 7, where Omega is A001222.
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EXAMPLE
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19 is in the sequence because 18 has 3 prime factors, 2, 3 and 3;
19 has 1 and 20 has 3 prime factors, 2, 2 and 5, for a total of 7 prime factors in the neighborhood.
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MATHEMATICA
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Select[Prime[Range[1000]], Total[FactorInteger[#^3 - #]][[2]] <= 7&] (* T. D. Noe, Apr 20 2011 *)
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PROG
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(PARI) isA106639(p)=my(g=gcd(p-1, 12)); isprime(p\g)&isprime((p+1)*g/24)&isprime(p) \\ Charles R Greathouse IV, Apr 20 2011
(PARI) forprime(p=1, 6000, if(bigomega(p-1)+bigomega(p+1)<=6, print1(p", "))) \\ Chris Boyd, Mar 23 2014
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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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