A172240 -id:A172240 - cp's OEIS frontend

A173176 Greater twin primes in A172240.

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

A172487 Lesser of twin primes in A172240.

Original entry on oeis.org

3, 17, 29, 41, 71, 101, 137, 149, 191, 197, 239, 269, 281, 311, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1427, 1451, 1481, 1607, 1667, 1697, 1721, 1787, 1871, 1877, 1931, 1949, 1997

Offset: 1

Giovanni Teofilatto, Nov 21 2010

For a(n) > 3, the first differences of the sequence are divisible by 6. (Is this a conjecture or a theorem?)

Cf. A173176, A172240, A181669, A181602.

isA001359 := 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:
isA181669 := proc(p) if isprime(p) and (p mod 6)= 5 then if numtheory[bigomega](p-1) =2 and  isA000430(p+2) then true; else false; end if;else false; end if ; end proc:
isA172240 := proc(n) isprime(n) and not isA181669(n) ; end proc:
isA172487 := proc(n) isA172240(n) and isA001359(n) ; end proc:
for n from 2 to 2000 do if isA172487(n) then printf("%d,",n) ; end if;end do:

A001359 INTERSECT A172240.

A172260 a(n) = A172240(n) + A172240(n+1).

Original entry on oeis.org

10, 20, 30, 36, 48, 60, 68, 78, 84, 96, 114, 128, 138, 144, 152, 162, 172, 186, 198, 204, 212, 222, 240, 258, 268, 276, 288, 300, 308, 320, 336, 354, 372, 384, 390, 396

Offset: 1

Giovanni Teofilatto, Nov 20 2010

For a(n) > 10, a(n) is divisible by 6 except if A172240(n+1) - A172240(n) is equal to 6.

Cf. A172240.

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

Original entry on oeis.org

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

A173641 Primes p such that p^2+4 and p^2-6 are both prime.

Original entry on oeis.org

3, 5, 7, 13, 17, 47, 67, 73, 97, 167, 193, 293, 317, 373, 463, 487, 503, 593, 607, 677, 787, 823, 827, 1087, 1613, 1637, 1987, 2477, 2543, 2687, 2777, 2833, 2903, 2957, 3023, 3583, 3593, 3917, 4093

Offset: 1

Giovanni Teofilatto, Nov 24 2010

p^2+4 and p^2-6 are both primes if a(n)^2+4 is in the sequence A172240 such that A172240(n) - 10 is also prime.

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

[p: p in PrimesUpTo(4100) | IsPrime(p^2+4)and IsPrime(p^2-6)]; // Vincenzo Librandi, Apr 16 2013

Select[Prime[Range[800]], PrimeQ[#^2 + 4]&& PrimeQ[#^2 - 6]&] (* Vincenzo Librandi, Apr 16 2013 *)

A173641 = list(p for p in primes(10^5) if is_prime(p^2+4) and is_prime(p^2-6))

Corrected and extended by D. S. McNeil, Nov 24 2010

cp's OEIS Frontend

A173176 Greater twin primes in A172240.

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A172487 Lesser of twin primes in A172240.

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A172260 a(n) = A172240(n) + A172240(n+1).

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

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A173641 Primes p such that p^2+4 and p^2-6 are both prime.

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