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A106639
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Distinguished primes.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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
| T. D. Noe, Table of n, a(n) for n = 1..1000
L. E. Dickson, A new extension of Dirichlet's theorem on prime numbers, Messenger of Math., 33 (1904), 155-161.
Walter Nissen, Home Page (listed in lieu of email address)
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FORMULA
| Primes p such that Omega(p^3 - p) <= 7, where Omega is A001222.
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EXAMPLE
| 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
| Select[Prime[Range[1000]], Total[FactorInteger[#^3 - #]][[2]] <= 7&] (* T. D. Noe, Apr 20 2011 *)
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PROG
| (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
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CROSSREFS
| Sequence in context: A049643 A050437 A096246 * A078334 A108696 A092581
Adjacent sequences: A106636 A106637 A106638 * A106640 A106641 A106642
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KEYWORD
| nonn
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AUTHOR
| Walter Nissen May 11 2005
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EXTENSIONS
| Formula, comment, offset, program, and link from Charles R Greathouse IV, Apr 20, 2011
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