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A350142
Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.
0
3, 5, 17, 65, 257, 65537, 4294967297
OFFSET
1,1
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.
EXAMPLE
For number 257 holds: tau(255) = 8, tau(256) = 9.
PROG
(Magma) [2^k + 1: k in [1..50] | #Divisors(2^k) - #Divisors(2^k-1) eq 1]
CROSSREFS
Intersection of (A055927+2) and A000051.
Sequence in context: A176964 A085749 A281623 * A278741 A265425 A256438
KEYWORD
nonn,hard,more
AUTHOR
Jaroslav Krizek, Dec 16 2021
STATUS
approved