A127072 Numbers k that divide 3^k - 2^k - 1.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 45, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Author

Alexander Adamchuk, Jan 04 2007

Keywords

• nonn

Comments

Prime p divides 3^p - 2^p - 1.

Quotients (3^p - 2^p - 1)/p, where p is prime, are listed in A127071.

Pseudoprimes in a(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073.

Numbers k such that k^2 divides 3^k - 2^k - 1 are listed in A127074.

Numbers k such that k^3 divides 3^k - 2^k - 1 are {1, 4, 7, ...}.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A127071, A127073, A127074.

Programs

Magma

[n: n in [1..250] | ((3^n - 2^n - 1) mod n) eq 0]; // G. C. Greubel, Aug 12 2019

Mathematica

Select[Range[1000],IntegerQ[(3^#-2^#-1)/# ]&]

PARI

is(n)=Mod(3,n)^n-Mod(2,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

Sage

[n for n in (1..250) if mod(3^n-2^n-1, n)==0 ] # G. C. Greubel, Jan 30 2020