A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

3, 5, 17, 65, 257, 65537, 4294967297

Offset: 1

Author

Jaroslav Krizek, Dec 16 2021

Keywords

• nonn
• hard
• more

Comments

Corresponding pairs of values [tau(m-2), tau(m-1)]: [1, 2], [2, 3], [4, 5], [6, 7], [8, 9], [16, 17], [32, 33], ...

There are no other terms <= 2^1206 + 1 (from A046801 data).

The first 5 known Fermat primes from A019434 are in this sequence. Corresponding values of tau(A019434(n - 2)): 1, 2, 4, 8, 16, ...

Conjecture 1: Also numbers m of the form 2^k + 1 such that tau(m - 2) = k.

Conjecture 2: If 6th Fermat prime F_p6 exists, then tau(F_p6 - 2) is a power of 2 and tau(F_p6 - 1) = tau(F_p6 - 2) + 1.

Conjecture 3: Sequence is finite with 7 terms; supersequence of A262534.

Examples

			For number 257 holds: tau(255) = 8, tau(256) = 9.
		

Crossrefs

Cf. A000005, A019434, A262534, A343144, A347078.

Intersection of (A055927+2) and A000051.

Programs

Magma

[2^k + 1: k in [1..50] | #Divisors(2^k) - #Divisors(2^k-1) eq 1];