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

1, 2, 3, 4, 7, 49, 179, 619, 17807

Offset: 1

Author

Alexander Adamchuk, Jan 04 2007

Keywords

• nonn
• hard
• more

Comments

No other terms below 10^9.

Prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p are listed in A127071.

Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.

The pseudoprimes in A127072 include all powers of primes and some composite numbers that are listed in A127073.

Numbers k such that k^3 divides 3^k - 2^k - 1 begin 1, 4, 7 (with no other terms < 10^8).

Primes in {a(n)} are {2,3,7,179,619,...}.

Crossrefs

Cf. A127071, A127072, A127073.

Programs

Magma

[n: n in [1..20000] | (3^n-2^n-1) mod n^2 eq 0]; // G. C. Greubel, Jan 30 2020

Mathematica

Do[f=(3^n-2^n-1);If[IntegerQ[f/n^2],Print[n]],{n,1,1000}]
Select[Range[20000], Mod[3^# -2^# -1, #^2]==0 &] (* G. C. Greubel, Jan 30 2020 *)

PARI

for(n=1, 20000, if((3^n-2^n-1)%n^2 == 0, print1(n", "))) \\ G. C. Greubel, Jan 30 2020

Sage

[n for n in (1..20000) if mod(3^n-2^n-1, n^2)==0 ] # G. C. Greubel, Jan 30 2020

Extensions

6 incorrect terms deleted by D. S. McNeil, Mar 16 2009 (the old version was 1,2,3,4,7,49,179,619,17807,95041,135433,393217,589825,1376257,1545601)

Edited by Max Alekseyev, Oct 21 2011