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

%I A242338 #30 Sep 08 2022 08:46:08
%S A242338 1,5,12,21,42,50,60,242,272
%N A242338 Numbers k such that k*7^k-1 is semiprime.
%C A242338 The semiprimes of this form are: 6, 84034, 166095446411, 11729463145748964146, 13102886255950779594655873516522994057, ...
%C A242338 From _Robert Israel_, Aug 19 2014: (Start)
%C A242338 If k is odd, k is in the sequence iff (k*7^k-1)/2 is prime.
%C A242338 If k == 1 (mod 3), k is in the sequence iff (k*7^k-1)/3 is prime.
%C A242338 a(10) >= 506.  506*7^506 - 1 is a 431-digit composite which apparently has not been factored.
%C A242338 (End)
%H A242338 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 A242338 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/cullen_woodall/7-.txt">Factorizations of n*7^n-1</a>
%p A242338 issemiprime:= proc(n) local F,t;
%p A242338     F:= ifactors(n,easy)[2];
%p A242338     t:= add(f[2],f=F);
%p A242338     if t = 1 then
%p A242338        if type(F[1][1],integer) then return false fi
%p A242338     elif t = 2 then
%p A242338        return not hastype(F,name)
%p A242338     else # t > 2
%p A242338        return false
%p A242338     fi;
%p A242338     F:= ifactors(n)[2];
%p A242338     return evalb(add(f[2],f=F)=2);
%p A242338 end proc:
%p A242338 select(n -> `if`(n::odd, isprime((n*7^n-1)/2),
%p A242338               issemiprime(n*7^n-1)), [$1..100]); # _Robert Israel_, Aug 19 2014
%t A242338 Select[Range[80], PrimeOmega[# 7^# - 1]==2&]
%o A242338 (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 A242338 (PARI) for(n=1,100,if(bigomega(n*7^n-1)==2,print1(n,", "))) \\ _Derek Orr_, Aug 20 2014
%Y A242338 Cf. similar sequences listed in A242273.
%Y A242338 Cf. A001358, A064753, A242200.
%K A242338 nonn,more
%O A242338 1,2
%A A242338 _Vincenzo Librandi_, May 12 2014
%E A242338 a(1) = 1 prepended and comment amended by _Harvey P. Dale_, Aug 12 2014
%E A242338 a(8) and a(9) from _Robert Israel_, Aug 20 2014