A259559 -id:A259559 - cp's OEIS frontend

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

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

nonn

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

			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.

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

Cf. A105403, A259559.

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

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

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

cp's OEIS Frontend

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

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