A063644 Primes with 2 representations: p*q*r - 1 = u*v*w + 1 where p, q, r, u, v and w are primes.

19, 29, 43, 67, 173, 283, 317, 653, 787, 907, 1867, 2083, 2693, 2803, 3413, 3643, 3677, 4253, 4363, 4723, 5443, 5717, 6197, 6547, 6653, 8563, 8573, 9067, 9187, 9403, 9643, 10733, 11443, 11587, 12163, 12917, 13997, 14107, 14683, 15187, 17827

Offset: 1

Author

Reinhard Zumkeller, Jul 21 2001

Keywords

• nonn

Comments

Also, primes sandwiched by 3-almost primes. Primes p such that p-+1 are 3-almost primes (A014612). - Zak Seidov, Jul 06 2015

Examples

			4723 is a term because 4723 = A063639(168)= 4724 - 1 = 2*2*1181 - 1, and because 4723 = A063640(158)= 4722 + 1 = 2*3*787 + 1.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Crossrefs

Cf. A063639, A063640, A014612.

Programs

Mathematica

Select[Prime[Range[3000]], 3 == PrimeOmega[# - 1] == PrimeOmega[# + 1] &] (* Vincenzo Librandi, Jul 07 2015 *)

PARI

n=0; default(primelimit, 2000000); for (m=2, 10^9, p=prime(m); if (bigomega(p + 1) == 3 && bigomega(p - 1) == 3, write("b063644.txt", n++, " ", p); if (n==1000, break)) ) \\ Harry J. Smith, Aug 27 2009

PARI

list(lim)=my(v=List(),u=v,L=(lim+1)\2,t); forprime(p=2,L\2, forprime(q=2,min(p,L\p), listput(u,p*q))); u=Set(u); for(i=2,#u, if(u[i]-u[i-1]==1 && isprime(t=2*u[i]-1), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jan 31 2017