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

3, 4, 5, 17, 257, 65537, 83623937

Offset: 1

Author

Jaroslav Krizek, Oct 27 2015

Keywords

• nonn
• more

Comments

Numbers k such that k = A000005(k) * A000010(k-2) + 1.

Sequence deviates from A249541; numbers 4294967297 and 6992962672132097 are not terms of this sequence.

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

Conjecture: primes from this sequence are in A254576.

a(8) > 10^25. If k = tau(k) * phi(k-2) + 1 then phi(k-2) must divide k-1, thus k-2 must be a term of A203966, which has already been searched up to 10^25. - Giovanni Resta, Feb 21 2020; updated by Max Alekseyev, Feb 21 2025

Examples

			17 is in this sequence because 17 = tau(17)*phi(15) + 1 = 2*8 + 1.
		

Crossrefs

Cf. A000005, A000010, A019434, A249541, A254576, A203966.

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

Programs

Magma

[n: n in [3..1000000] |  n eq NumberOfDivisors(n) * EulerPhi(n-2) + 1];

Mathematica

Select[Range@ 100000, # == DivisorSigma[0, #] EulerPhi[# - 2] + 1 &] (* Michael De Vlieger, Oct 27 2015 *)

PARI

for(n=3, 1e8, if(numdiv(n)*eulerphi(n-2) == n-1, print1(n ", "))) \\ Altug Alkan, Oct 28 2015

PARI

lista(na, nb) = {my(f1 = factor(na-2), f2 = factor(na-1), f3); for(n=na, nb, f3 = factor(n); if (numdiv(f3)*eulerphi(f1) == n-1, print1(n ", ")); f1 = f2; f2 = f3;);}; \\ Michel Marcus, Feb 21 2020