A242338 Numbers k such that k*7^k-1 is semiprime.

1, 5, 12, 21, 42, 50, 60, 242, 272

Offset: 1

Author

Vincenzo Librandi, May 12 2014

Keywords

• nonn
• more

Comments

The semiprimes of this form are: 6, 84034, 166095446411, 11729463145748964146, 13102886255950779594655873516522994057, ...

From Robert Israel, Aug 19 2014: (Start)

If k is odd, k is in the sequence iff (k*7^k-1)/2 is prime.

If k == 1 (mod 3), k is in the sequence iff (k*7^k-1)/3 is prime.

a(10) >= 506. 506*7^506 - 1 is a 431-digit composite which apparently has not been factored.

(End)

Links

• Paul Leyland, Cullen and Woodall numbers and their generalization to other bases
• Paul Leyland, Factorizations of n*7^n-1

Crossrefs

Cf. similar sequences listed in A242273.

Cf. A001358, A064753, A242200.

Programs

Magma

IsSemiprime:=func; [n: n in [2..80] | IsSemiprime(s) where s is n*7^n-1];

Maple

issemiprime:= proc(n) local F,t;
    F:= ifactors(n,easy)[2];
    t:= add(f[2],f=F);
    if t = 1 then
       if type(F[1][1],integer) then return false fi
    elif t = 2 then
       return not hastype(F,name)
    else # t > 2
       return false
    fi;
    F:= ifactors(n)[2];
    return evalb(add(f[2],f=F)=2);
end proc:
select(n -> `if`(n::odd, isprime((n*7^n-1)/2),
              issemiprime(n*7^n-1)), [$1..100]); # Robert Israel, Aug 19 2014

Mathematica

Select[Range[80], PrimeOmega[# 7^# - 1]==2&]

PARI

for(n=1,100,if(bigomega(n*7^n-1)==2,print1(n,", "))) \\ Derek Orr, Aug 20 2014

Extensions

a(1) = 1 prepended and comment amended by Harvey P. Dale, Aug 12 2014

a(8) and a(9) from Robert Israel, Aug 20 2014