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A181602 Primes p such that p-1 is a semiprime and p+2 is prime or prime squared. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

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

MATHEMATICA

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 *)

PROG

(MAGMA) [ 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

CROSSREFS

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

Cf. A173176, A172487, A172240, A181669.

Sequence in context: A090810 A092307 A005385 * A075705 A249735 A218394

Adjacent sequences:  A181599 A181600 A181601 * A181603 A181604 A181605

KEYWORD

nonn

AUTHOR

Giovanni Teofilatto, Nov 01 2010

EXTENSIONS

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

STATUS

approved

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Last modified October 19 14:36 EDT 2019. Contains 328222 sequences. (Running on oeis4.)