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

1,1

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.

Links

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Sequence in context: A033236 A260808 A141457 * A339778 A328160 A118162

Adjacent sequences: A112995 A112996 A112997 * A112999 A113000 A113001

Keyword

nonn

Author

Zak Seidov, Jan 03 2006

Extensions

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

Status

approved