%I #30 Sep 08 2022 08:46:08

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

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

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

%C From _Robert Israel_, Aug 19 2014: (Start)

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

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

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

%C (End)

%H Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/cw.html">Cullen and Woodall numbers and their generalization to other bases</a>

%H Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/7-.txt">Factorizations of n*7^n-1</a>

%p issemiprime:= proc(n) local F,t;

%p F:= ifactors(n,easy)[2];

%p t:= add(f[2],f=F);

%p if t = 1 then

%p if type(F[1][1],integer) then return false fi

%p elif t = 2 then

%p return not hastype(F,name)

%p else # t > 2

%p return false

%p fi;

%p F:= ifactors(n)[2];

%p return evalb(add(f[2],f=F)=2);

%p end proc:

%p select(n -> `if`(n::odd, isprime((n*7^n-1)/2),

%p issemiprime(n*7^n-1)), [$1..100]); # _Robert Israel_, Aug 19 2014

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

%o (Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [2..80] | IsSemiprime(s) where s is n*7^n-1];

%o (PARI) for(n=1,100,if(bigomega(n*7^n-1)==2,print1(n,", "))) \\ _Derek Orr_, Aug 20 2014

%Y Cf. similar sequences listed in A242273.

%Y Cf. A001358, A064753, A242200.

%K nonn,more

%O 1,2

%A _Vincenzo Librandi_, May 12 2014

%E a(1) = 1 prepended and comment amended by _Harvey P. Dale_, Aug 12 2014

%E a(8) and a(9) from _Robert Israel_, Aug 20 2014