A112998 Numbers n such that n, n+1 and n+2 are 1,2,3-almost primes.

61, 73, 193, 277, 397, 421, 613, 661, 757, 1093, 1237, 1453, 1657, 2137, 2341, 2593, 2797, 2917, 3217, 4177, 4621, 5233, 6121, 6133, 6217, 7057, 7537, 8101, 8317, 8353, 8521, 8677, 8893, 9013, 9277, 9721, 9817, 10357, 10957, 11161, 11677, 11701, 12301

Offset: 1

Author

Zak Seidov, Jan 03 2006

Keywords

• nonn

Comments

It's easy to see that all terms = 1 modulo 12. Primes p are {1,5,7,11} mod 12. Then p+1 = {2,6,8,12} mod 12 and only 2 mod 12 may give semiprime. Cf. A186696 for values of (a(n)-1)/12 = 5,6,16,23,33,35, etc.

Examples

			61 is prime, 62=2*31 is semiprime, 63=3*3*7 is 3-almost prime.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000

Programs

Mathematica

Do[p=Prime[n];If[Total[FactorInteger[p+1]][[2]]==2&&Total[FactorInteger[p+2]][[2]]==3, Print[p]], {n, 1, 1000}];

PARI

list(lim)=my(v=List(),L=(lim+2)\3,t); forprime(p=3,L\3, forprime(q=3,min(L\p,p), t=3*p*q-2; if(t%12==1 && isprime(t) && isprime((t+1)/2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Extensions

Extended and edited by Charles R Greathouse IV, Apr 20 2010