A347078 -id:A347078 - cp's OEIS frontend

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

Jaroslav Krizek, Dec 16 2021

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.

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

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

Intersection of (A055927+2) and A000051.

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

A351890 Primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3), where tau(k) is the number of divisors of k (A000005).

Original entry on oeis.org

5, 17, 65537, 9632244737, 20892967937, 127831991297, 149255504897, 159667373057, 351108391937, 542497063937, 1650957730817, 2270398022657, 2322380932097, 2747956028417, 2888694547457, 3516735087617, 6029264167937, 6122338640897, 6705696695297, 11125266727937

Offset: 1

Jaroslav Krizek, Mar 03 2022

nonn

Corresponding values of tau(a(n)-1): 3, 5, 17, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, ...
Corresponding values of tau(a(n)-2) = tau(a(n)-3): 2, 4, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, ...
Quadruples of [tau(a(n)-3), tau(a(n)-2), tau(a(n)-1), tau(a(n))]: [2, 2, 3, 2], [4, 4, 5, 2], [16, 16, 17, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], ...
Quadruple [32, 32, 33, 2] holds for all 128 terms 65537 < a(n) < 10^15.
Number p-1 is a perfect square as its number of divisors is odd.
The first 3 terms are Fermat primes from A019434.
Term 103565955613697 is the smallest primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3) = tau(p - 4).

			Quadruple of [tau(65534), tau(65535), tau(65536), tau(65537)]: [16, 16, 17, 2].

Subsequence of A347078.

Cf. A000005 (tau), A019434.

[m: m in [4..10^6] | IsPrime(m) and #Divisors(m - 1) eq #Divisors(m - 2) + 1 and #Divisors(m - 2) eq #Divisors(m - 3)];

cp's OEIS Frontend

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

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