A263811 Numbers k such that k = tau(k) * phi(k-1) + 1.

3, 5, 17, 25, 49, 257, 289, 65537

Offset

1,1

Comments

Numbers k such that k = A000005(k) * A000010(k-1) + 1.

The first 5 known Fermat primes from A019434 are in this sequence.

The next term, if it exists, must be greater than 2*10^7.

A prime p is in the sequence iff p is a Fermat prime (A019434) - see proof in A171271.

Observation: the known composite terms are squares of primes. - Omar E. Pol, Nov 04 2015

From Charlie Neder, Mar 02 2019: (Start)

Rearranging the definition gives (k-1)/phi(k-1) = tau(k), which means k-1 is in A007694. Since k-1 is thus 3-smooth, there are two possibilities:

1) k-1 is a power of 2 and tau(k) = 2, i.e., k is a Fermat prime,

2) k-1 is a 3-smooth number divisible by 6 and tau(k) = 3, i.e., k is a Pierpont number and the square of a prime.

In the second case, k-1 factors as (p-1)(p+1) for some p, and both parts are 3-smooth if and only if p is in {2,3,5,7,17} (2 and 3 are excluded since in those cases k-1 is not divisible by 6). Therefore, this sequence is complete if and only if there are no more Fermat primes. (End)

Links

Table of n, a(n) for n=1..8.

Example

17 is in this sequence because 17 = tau(17)*phi(16) + 1 = 2*8 + 1.

Mathematica

Select[Range[10^5], # == DivisorSigma[0, #] EulerPhi[# - 1] + 1 &] (* Michael De Vlieger, Nov 05 2015 *)

Prog

(Magma) [n: n in [2..1000000] |  n eq NumberOfDivisors(n) * EulerPhi(n-1) + 1]
(PARI) for(n=1, 1e5, if( n-1 == numdiv(n)*eulerphi(n-1) , print1(n, ", "))) \\ Altug Alkan, Nov 05 2015

Crossrefs

Cf. A000005, A000010, A019434.

Cf. A263810 (numbers k such that k = tau(k) * phi(k-2) + 1).

Sequence in context: A354724 A255401 A364959 * A323194 A024867 A025111

Adjacent sequences: A263808 A263809 A263810 * A263812 A263813 A263814

Keyword

nonn,hard,more

Author

Jaroslav Krizek, Nov 04 2015

Status

approved