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).

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

Offset

1,1

Comments

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).

Links

Table of n, a(n) for n=1..20.

Example

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

Prog

(Magma) [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)]

Crossrefs

Subsequence of A347078.

Cf. A000005 (tau), A019434.

Sequence in context: A062223 A363759 A116911 * A097491 A120087 A353690

Adjacent sequences: A351887 A351888 A351889 * A351891 A351892 A351893

Keyword

nonn

Author

Jaroslav Krizek, Mar 03 2022

Status

approved