A172487 -id:A172487 - cp's OEIS frontend

A181602 Primes p such that p-1 is a semiprime and p+2 is prime or prime squared.

5, 7, 11, 23, 47, 59, 107, 167, 179, 227, 347, 359, 839, 1019, 1319, 1367, 1487, 1619, 2027, 2207, 2999, 3119, 3167, 3467, 4127, 4259, 4547, 4787, 4799, 5099, 5639, 5879, 6659, 6779, 6827, 7559, 8819, 10007, 10607, 11699, 12107, 12539, 14387, 14867

Offset: 1

Giovanni Teofilatto, Nov 01 2010

nonn

Except for the second term, a(n)+1 is divisible by 6.
[Proof: a(n)=p is a prime, with p-1=q*r and two primes q<=r by definition. Omitting the special case p=2, p is odd, p+1 is even, so p+1=q*r+2 = 2(1+q*r/2). To show that p+1 is divisible by 6 we show that it is divisible by 2 and by 3; divisibility by 2 has already been shown in the previous sentence. (1+q*r/2 must be integer, so q*r/2 must be integer, so the smaller prime q of the semiprime must be q=2, so p=2*r+1. This shows that p=a(n) are a subset of A005383.) First subcase of the definition is that p+2 is also prime. Then p is a smaller twin prime and by a comment in A003627, p+1 is divisible by 3. Second subcase of the definition is that p+2 = s^2 with s a prime. s can be 3*k+1 or 3*k+2 --p=7 is the exception-- which leads to s^2 = 9*k^2+6*k+1 or s^2=9*k^2+12*k+4, so p+1 = 9*k^2+6*k or 9*k^2+12*k+3, and in both cases p+1 is divisible by 3.]
In consequence, except for the first three terms, first differences a(n+1)-a(n) are also divisible by 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A001248 (squares of primes).

Cf. A173176, A172487, A172240, A181669.

[ p: p in PrimesInInterval(3,15000) | &+[ k[2]: k in Factorization(p-1) ] eq 2 and (IsPrime(p+2) or (q^2 eq p+2 and IsPrime(q) where q is Isqrt(p+2))) ]; // Klaus Brockhaus, Nov 03 2010

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; fQ[n_] := Block[{fi = FactorInteger@n}, Length@ fi == 1 && fi[[1, 2]] == 1 || fi[[1, 2]] == 2]; Select[ Prime@ Range@ 1293, semiPrimeQ[ # - 1] && fQ[ # + 2] &] (* Robert G. Wilson v, Nov 06 2010 *)
Select[Prime[Range[2000]],PrimeOmega[#-1]==2&&Or@@PrimeQ[{#+2, Sqrt[ #+2]}]&] (* Harvey P. Dale, Aug 12 2012 *)

Corrected (29 removed) and extended by Klaus Brockhaus, Robert G. Wilson v and R. J. Mathar, Nov 03 2010

A172240 Odd primes not in A181669.

Original entry on oeis.org

3, 7, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Giovanni Teofilatto, Nov 20 2010

Except for term 5, the sequence contains all greater of twin primes

Cf. A001359, A173176, A172487, A181669, A181602, A173641.

A173176 Greater twin primes in A172240.

Original entry on oeis.org

7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609, 1621, 1669, 1699, 1723, 1789, 1873, 1879, 1933, 1951, 1999

Offset: 1

Giovanni Teofilatto, Nov 22 2010

For a(n) > 5, first difference of the sequence is divisible by 6. (Conjectured or proved?)

Also for a(n)>5, a(n)-1 is divisible by 6, if a(n)-2 is prime p such that p+1 is divisible by 6.

Cf. A172487, A181669, A181602.

isA006512 := proc(p) isprime(p) and isprime(p-2) ; end proc:
isA000430 := proc(p) if isprime(p) then true; else if issqr(p) then isprime(sqrt(p)) ; else false; end if; end if; end proc:
isA181602 := proc(p) if isprime(p) then if numtheory[bigomega](p-1) =2 and  isA000430(p+2) then true; else false; end if; else false;   end if ; end proc:
isA181669 := proc(p) isA181602(p) and (p mod 6)= 5 ; end proc:
isA172240 := proc(n) isprime(n) and not isA181669(n) ; end proc:
isA173176 := proc(n) isA172240(n) and isA006512(n) ; end proc:
for n from 2 to 2000 do if isA173176(n) then printf("%d,",n) ; end if; end do:

A172240 INTERSECT A006512.

Corrected by R. J. Mathar, Dec 01 2010

cp's OEIS Frontend

A181602 Primes p such that p-1 is a semiprime and p+2 is prime or prime squared.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A172240 Odd primes not in A181669.

Views

Author

Keywords

Comments

Crossrefs

A173176 Greater twin primes in A172240.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

Extensions