A259558 Numbers n such that prime(n)-1 and prime(n+1)-1 have the same number of distinct prime factors.

2, 4, 5, 8, 9, 12, 15, 16, 18, 19, 23, 24, 25, 28, 29, 31, 36, 38, 39, 40, 42, 44, 52, 56, 58, 59, 60, 63, 64, 71, 73, 74, 76, 80, 85, 88, 91, 92, 94, 96, 98, 99, 102, 103, 106, 107, 109, 110, 111, 112, 113, 117, 126, 129, 130, 131, 132, 133, 134, 136, 139, 141, 142, 143, 144, 151, 152, 159, 160, 161, 165, 168, 169, 173

Offset

1,1

Comments

Unlike A105403, this sequence appears to be infinite.

Dickson's conjecture would imply that there are infinitely many p such that p, p+6, 2*p+1 and 2*p+13 are prime and there are no primes between 2*p+1 and 2*p+13. Then n is in the sequence where 2*p+1=prime(n). - Robert Israel, Jun 30 2015

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The prime factors of prime(5)-1 are 2,5. The prime factors of prime(6)-1 are 2,3,3 and they have the same number of distinct prime factors.

Maple

N:= 2000: # to use primes <= N
Primes:= select(isprime, [2, seq(2*i+1, i=1..floor((N-1)/2))]):
npf:= map(t -> nops(numtheory:-factorset(Primes[t]-1)), [$1..nops(Primes)]):
select(t -> npf[t+1]=npf[t], [$1..nops(Primes)-1]); # Robert Israel, Jun 30 2015

Mathematica

Select[Range@ 173, PrimeNu[Prime[#] - 1] == PrimeNu[Prime[# + 1] - 1] &] (* Michael De Vlieger, Jul 01 2015 *)

Prog

(PARI) lista(nn) = {forprime(p=2, nn, if (omega(p-1)==omega(nextprime(p+1)-1), print1(primepi(p), ", ")); ); } \\ Michel Marcus, Jul 01 2015

Crossrefs

Cf. A105403, A259559.

Sequence in context: A239953 A321324 A343013 * A352778 A189140 A189134

Adjacent sequences: A259555 A259556 A259557 * A259559 A259560 A259561

Keyword

nonn

Author

Pratik Koirala, Otis Tweneboah, Nathan Fox, Eugene Fiorini, Jun 30 2015

Status

approved