A296104 Numbers k such that 2^k == 3 (mod k-1).

2, 111482, 465794, 79036178, 1781269903308, 250369632905748, 708229497085910, 15673900819204068

Offset: 1

Author

Krzysztof Ziemak and Max Alekseyev, Dec 04 2017

Keywords

• nonn
• more

Comments

Also, numbers k such that 2^k - 2 is a Fermat pseudoprime, i.e., 2^k - 2 belongs to A015919 and A006935.

a(3) was found by McDaniel (1989).

Some larger terms (maybe not in order): 2338990834231272653582, 341569682872976768698011746141903924998969680638.

Discovered huge even PSP(2) numbers of the form 2*M(n), where n=p*q and M(n)=2^n-1, ensure that the following numbers are also even pseudoprimes of the form 2*M(p)*M(q): 2*M(37)*M(12589), 2*M(131)*M(17854891864360859951), 2*M(179)*M(1398713032993), 2*M(2111)*M(335494787819), 2*M(35267)*M(50508121). - Krzysztof Ziemak, Jan 01 2018

Links

• W. L. McDaniel, Some Pseudoprimes and Related Numbers Having Special Forms, Math. Comp. 53:187 (1989), 407-409.

Crossrefs

Cf. A000079, A015919, A006935, A014741, A050259, A055685, A296370.

Programs

Mathematica

k = 2; lst = {2}; While[k < 1000000001, If[ PowerMod[2, k, k -1] == 3, AppendTo[lst, k]]; k += 10; If[ PowerMod[2, k, k -1] == 3, AppendTo[lst, k]]; k += 2]; lst (* Robert G. Wilson v, Jan 01 2018 *)

PARI

is_A296104(n) = Mod(2, n-1)^n == 3; \\ Iain Fox, Dec 07 2017

Python

A296104_list = [n for n in range(2,10**6) if pow(2,n,n-1) == 3 % (n-1)] # Chai Wah Wu, Dec 06 2017

Formula

a(n) = A296370(n) + 1.