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

2, 3, 4, 6, 8, 10, 15, 16, 22, 36, 46, 58, 82, 100, 106, 120, 166, 168, 178, 196, 210, 226, 256, 262, 270, 280, 312, 330, 346, 358, 378, 382, 408, 456, 462, 466, 478, 502, 520, 546, 562, 570, 586, 616, 640, 676, 690, 718, 728, 750, 760, 838, 858, 862, 886

Offset: 1

Author

Claude H. R. Dequatre, Apr 17 2021

Keywords

• nonn
• easy

Comments

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).

Conjecture: all members except 2 and 3 are composite.

Open question: is there a finite number of odd terms in this sequence?

Examples

			16 is a term because tau(16) = 5 and tau(5) = 2 and tau(17) = 2 and 2 is prime.
23 is not a term because tau(23) = 2 and tau(2) = 2 and tau(24) = 8 and 2 is prime but not 8.
98 is not a term because tau(98) = 6 and tau(6) = 4 and tau(99) = 6 and 4 and 6 are not prime.
		

Links

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

Crossrefs

Cf. A000005, A000040, A010553. Includes A077065.

Programs

Maple

filter:= proc(n)
  isprime(numtheory:-tau(n+1)) and isprime(numtheory:-tau(numtheory:-tau(n)))
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2025

Mathematica

With[{t = DivisorSigma}, Select[Range[1000], And @@ PrimeQ[{t[0, t[0, #]], t[0, # + 1]}] &]] (* Amiram Eldar, May 27 2021 *)

PARI

for(k=1,1e4,if(isprime(numdiv(numdiv(k))) && isprime(numdiv(k+1)),print1(k", ")))