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A259558 Numbers n such that prime(n)-1 and prime(n+1)-1 have the same number of distinct prime factors. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A002541 A239953 A321324 * A189140 A189134 A189019

Adjacent sequences:  A259555 A259556 A259557 * A259559 A259560 A259561

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified September 18 07:58 EDT 2019. Contains 327168 sequences. (Running on oeis4.)