A164290 -id:A164290 - cp's OEIS frontend

A164289 Sequence of twin primes p where the middle term p+1 has 5 prime factors (here p+2 is the associated twin prime, not listed).

71, 107, 179, 269, 311, 419, 659, 827, 881, 1019, 1031, 1049, 1091, 1427, 1451, 1607, 1931, 1949, 1997, 2027, 2141, 2309, 2549, 2711, 2729, 2789, 3329, 3467, 3539, 3767, 3821, 3851, 4019, 4091, 4229, 4259, 4481, 4649, 4931, 5417, 5651, 5741, 5867, 6089

Offset: 1

Carlos Alves, Aug 12 2009

nonn

This sequence is similar to: A060213, with 3 prime factors in the middle number; A102168, with 4 prime factors in the middle number.

These sequences are of the form (p,p+1,p+2) with (p,p+2) twin primes and Omega(p+1)=m with m>=3 (m=1 or m=2 is impossible).

			71 is a term since 71 and 73 are twin primes and Omega(71 + 1) = Omega(72) = Omega(2*2*2*3*3) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A060213, A102168, A164290.

Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 30000]]; Asequence = Flatten@Position[Partition[Wmil, 3, 1], {1, 5, 1}]
Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==2&&PrimeOmega[ #[[1]]+1] ==5&][[All,1]] (* Harvey P. Dale, Apr 02 2022 *)

A164291 a(n) = p is the first twin prime (p, p+2) for which p+1 has n prime factors (n>=2, multiplicity counted).

Original entry on oeis.org

3, 11, 59, 71, 239, 191, 2111, 1151, 14591, 26111, 15359, 139967, 138239, 675839, 2101247, 737279, 4866047, 786431, 22118399, 36175871, 194641919, 63700991, 138412031, 169869311, 1321205759, 11123294207, 16357785599, 4076863487, 25165823999, 10871635967

Offset: 2

Carlos Alves, Aug 12 2009

nonn

a(3)-a(6) are the first elements of A060213, A102168, A164289, A164290 respectively with n=3,4,5,6 (prime factors in the middle number).

This gives the first p with (p,p+2) twin primes and Omega(p+1)=n with n>=2 (n=1 is impossible).

			a(7)=191 because in (191, 192, 193) we have Omega(192)=Omega(2*2*2*2*2*2*3)=7 and 191, 193 are twin primes.
The sequence oscillates and here we see that a(7)<a(6)=239.

Donovan Johnson, Table of n, a(n) for n = 2..200

Cf. A060213, A102168, A164289, A164290.

Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 10000000]]; Aseq=(Flatten@Position[Partition[Wmil, 3, 1], {1, #, 1}])[[1]] & /@ Range[3,19]

Definition and comments corrected, a(2) and a(20)-a(29) from Donovan Johnson, Aug 20 2009

cp's OEIS Frontend

A164289 Sequence of twin primes p where the middle term p+1 has 5 prime factors (here p+2 is the associated twin prime, not listed).

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