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%I #19 Jun 13 2021 15:27:12
%S 2,3,4,6,8,10,15,16,22,36,46,58,82,100,106,120,166,168,178,196,210,
%T 226,256,262,270,280,312,330,346,358,378,382,408,456,462,466,478,502,
%U 520,546,562,570,586,616,640,676,690,718,728,750,760,838,858,862,886
%N Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.
%C 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 Conjecture: all members except 2 and 3 are composite.
%C Open question: is there a finite number of odd terms in this sequence?
%e 16 is a term because tau(16) = 5 and tau(5) = 2 and tau(17) = 2 and 2 is prime.
%e 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 98 is not a term because tau(98) = 6 and tau(6) = 4 and tau(99) = 6 and 4 and 6 are not prime.
%t With[{t = DivisorSigma}, Select[Range[1000], And @@ PrimeQ[{t[0, t[0, #]], t[0, # + 1]}] &]] (* _Amiram Eldar_, May 27 2021 *)
%o (PARI) for(k=1,1e4,if(isprime(numdiv(numdiv(k))) && isprime(numdiv(k+1)),print1(k", ")))
%Y Cf. A000005, A000040, A010553.
%K nonn,easy
%O 1,1
%A _Claude H. R. Dequatre_, Apr 17 2021
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