cp's OEIS Frontend

A343502 Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.

%I A343502 #27 Feb 03 2025 01:59:18
%S A343502 2,3,4,6,8,10,15,16,22,36,46,58,82,100,106,120,166,168,178,196,210,
%T A343502 226,256,262,270,280,312,330,346,358,378,382,408,456,462,466,478,502,
%U A343502 520,546,562,570,586,616,640,676,690,718,728,750,760,838,858,862,886
%N A343502 Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.
%C A343502 Considering the first 10^8 positive integers there are 1439855 terms in the sequence and only the first two (2,3) are prime, all the others are composite numbers of which only three are odd (15, 65535 and 4194303).
%C A343502 Conjecture: all members except 2 and 3 are composite.
%C A343502 Open question: is there a finite number of odd terms in this sequence?
%H A343502 Robert Israel, <a href="/A343502/b343502.txt">Table of n, a(n) for n = 1..10000</a>
%e A343502 16 is a term because tau(16) = 5 and tau(5) = 2 and tau(17) = 2 and 2 is prime.
%e A343502 23 is not a term because tau(23) = 2 and tau(2) = 2 and tau(24) = 8 and 2 is prime but not 8.
%e A343502 98 is not a term because tau(98) = 6 and tau(6) = 4 and tau(99) = 6 and 4 and 6 are not prime.
%p A343502 filter:= proc(n)
%p A343502   isprime(numtheory:-tau(n+1)) and isprime(numtheory:-tau(numtheory:-tau(n)))
%p A343502 end proc:
%p A343502 select(filter, [$1..1000]); # _Robert Israel_, Feb 02 2025
%t A343502 With[{t = DivisorSigma}, Select[Range[1000], And @@ PrimeQ[{t[0, t[0, #]], t[0, # + 1]}] &]] (* _Amiram Eldar_, May 27 2021 *)
%o A343502 (PARI) for(k=1,1e4,if(isprime(numdiv(numdiv(k))) && isprime(numdiv(k+1)),print1(k", ")))
%Y A343502 Cf. A000005, A000040, A010553. Includes A077065.
%K A343502 nonn,easy
%O A343502 1,1
%A A343502 _Claude H. R. Dequatre_, Apr 17 2021