A252659 Numbers m such that 6^m - m is a semiprime.

2, 3, 5, 10, 15, 23, 34, 37, 47, 70, 259, 275, 278, 497, 563

Offset: 1

Author

Vincenzo Librandi, Dec 21 2014

Keywords

• nonn
• more
• hard

Comments

From Robert Israel, Sep 06 2016: (Start)

Even n is in this sequence iff (6^n-n)/2 is prime.

3*k is in this sequence iff 2*6^(3*k-1)-k is prime.

Also contains 275, 278 and 683.

The only other possible member less than 275 is 259. (End)

a(16) >= 617. - Kevin P. Thompson, Apr 29 2022

Examples

			2 is in this sequence because 6^2-2 = 2*17 is semiprime.
10 is in this sequence because 6^10-10 = 2*30233083 and these two factors are prime.
		

Links

• FactorDB, Status of 6^617-617

Crossrefs

Cf. similar sequences listed in A252656.

Programs

Magma

IsSemiprime:=func; [m: m in [1..90] | IsSemiprime(s) where s is 6^m-m];

Maple

Res:= NULL:
for n from 1 to 100 do
    F:= ifactors(6^n-n, easy)[2];
    if add(t[2], t=F) >= 3 or (hastype(F, symbol) and add(t[2], t=F) >= 2)
       then flag:= false
    elif add(t[2], t=F) = 2 and not hastype(F, symbol) then flag:= true
    else
     flag:= evalb(numtheory:-bigomega(6^n-n)=2)
    fi;
  if flag then  Res:= Res, n fi
od:
Res; # Robert Israel, Sep 06 2016

Mathematica

Select[Range[90], PrimeOmega[6^# - #]== 2&]

Extensions

a(11)-a(15) from Kevin P. Thompson, Apr 29 2022