A239669 -id:A239669 - cp's OEIS frontend

A106639 Distinguished primes.

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

Offset: 1

Walter Nissen, May 11 2005

nonn

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

			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.

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.

Cf. A239669.

Select[Prime[Range[1000]], Total[FactorInteger[#^3 - #]][[2]] <= 7&] (* T. D. Noe, Apr 20 2011 *)

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

forprime(p=1,6000,if(bigomega(p-1)+bigomega(p+1)<=6,print1(p", "))) \\ Chris Boyd, Mar 23 2014

Primes p such that Omega(p^3 - p) <= 7, where Omega is A001222.

Formula, comment, offset, program, and link from Charles R Greathouse IV, Apr 20 2011

A155800 a(n) = smallest prime p such that p-1 and p+1 together have n prime divisors, or a(n) = 0 if no such prime exists.

Original entry on oeis.org

2, 0, 3, 5, 7, 19, 17, 31, 97, 191, 127, 449, 769, 3329, 6143, 7937, 12799, 51199, 8191, 165887, 65537, 131071, 1179649, 2654209, 7995391, 524287, 10616831, 12910591, 167772161, 113246209, 169869311, 155189249, 1887436799, 3221225473

Offset: 1

J. M. Bergot, Jan 27 2009

nonn

Prime divisors are counted with multiplicity.

			For p=2, the adjacent numbers 1 and 3 together have one prime divisor, hence a(1) = 2. For p=3, the adjacent numbers 2 and 4 together have three prime divisors, hence a(3) = 3. For primes greater than 3, the adjacent numbers are composite and therefore together have at least four prime divisors, so no prime exists whose neighbors together have two prime divisors. Hence a(2) = 0.
For p = 19, p-1 = 18 = 2*3*3 and p+1 = 20 = 2*2*5 together have six prime divisors. All smaller primes are surrounded by numbers which together have fewer or more than six (for 17 there are seven) prime divisors, hence a(6) = 19.

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..46 (terms < 10^13, first 40 terms from Donovan Johnson)

Cf. A001222 (number of prime divisors of n), A154598, A155850, A154704.

Cf. comments in A239669.

Join[{2,0},With[{m=SortBy[{PrimeOmega[#-1]+PrimeOmega[#+1],#}&/@Prime[ Range[200000]],First]},Transpose[Flatten[Table[Select[m,#[[1]]==n&,1],{n,3,24}],1]][[2]]]] (* Harvey P. Dale, Sep 24 2013 *)

N=10^7; default(primelimit, N); M=26;
/* M must be determined empirically for each N. Program-generated zeros other than a(2) appearing as terms when N and M are increased must be regarded as provisional */
for(m=1,M,t=0;forprime(p=1,N,if(bigomega(p-1)+bigomega(p+1)==m,t=1;print1(p", ");break));if(t==0,print1("0, "))) \\ Chris Boyd, Mar 24 2014

Edited, 1151 replaced by 769, 3457 replaced by 3329, extended beyond a(14) by Klaus Brockhaus, Jan 28 2009
a(29)-a(32) from Klaus Brockhaus, Jan 30 2009
a(33)-a(34) from Donovan Johnson, Aug 03 2009

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A106639 Distinguished primes.

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A155800 a(n) = smallest prime p such that p-1 and p+1 together have n prime divisors, or a(n) = 0 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions