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

1,1

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?

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000005, A000040, A010553. Includes A077065.

Sequence in context: A323360 A375186 A297417 * A238876 A211856 A066816

Adjacent sequences: A343499 A343500 A343501 * A343503 A343504 A343505

Keyword

nonn,easy

Author

Claude H. R. Dequatre, Apr 17 2021

Status

approved