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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

%I #24 Jul 07 2015 15:12:55

%S 2,4,5,8,9,12,15,16,18,19,23,24,25,28,29,31,36,38,39,40,42,44,52,56,

%T 58,59,60,63,64,71,73,74,76,80,85,88,91,92,94,96,98,99,102,103,106,

%U 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

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

%C Unlike A105403, this sequence appears to be infinite.

%C 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

%H Robert Israel, <a href="/A259558/b259558.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%p N:= 2000: # to use primes <= N

%p Primes:= select(isprime,[2,seq(2*i+1,i=1..floor((N-1)/2))]):

%p npf:= map(t -> nops(numtheory:-factorset(Primes[t]-1)), [$1..nops(Primes)]):

%p select(t -> npf[t+1]=npf[t],[$1..nops(Primes)-1]); # _Robert Israel_, Jun 30 2015

%t Select[Range@ 173, PrimeNu[Prime[#] - 1] == PrimeNu[Prime[# + 1] - 1] &] (* _Michael De Vlieger_, Jul 01 2015 *)

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

%Y Cf. A105403, A259559.

%K nonn

%O 1,1

%A _Pratik Koirala_, _Otis Tweneboah_, _Nathan Fox_, _Eugene Fiorini_, Jun 30 2015

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Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)