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Semiprimes that are the concatenation of two or more semiprimes.
1

%I #15 Aug 24 2023 10:16:48

%S 46,49,69,94,106,146,159,214,219,226,254,259,334,339,346,386,394,415,

%T 422,446,451,458,466,469,482,485,493,514,519,554,559,579,586,589,614,

%U 622,626,629,633,634,635,649,655,662,669,674,685,687,694,695,699,746,749,779,866,869,879,914,921,922

%N Semiprimes that are the concatenation of two or more semiprimes.

%C Conjecture: The fraction of semiprimes <= N that are in this sequence goes to 1 as N -> infinity. What is the first N for which that fraction >= 1/2?

%H Robert Israel, <a href="/A365009/b365009.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 69 is a term because 69 = 3 * 23 is a semiprime and is the concatenation of the semiprimes 6 = 2 * 3 and 9 = 3 * 3.

%p filter:= proc(n) local d,v;

%p if numtheory:-bigomega(n) <> 2 then return false fi;

%p for d from 1 to length(n)-1 do

%p v:= n mod 10^d;

%p if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and g((n-v)/10^d) then return true fi

%p od;

%p false

%p end proc:

%p g:= proc(n) local d,v; option remember;

%p if numtheory:-bigomega(n) = 2 then return true fi;

%p for d from 1 to length(n)-1 do

%p v:= n mod 10^d;

%p if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and procname((n-v)/10^d) then return true fi

%p od;

%p false

%p end proc:

%p select(filter, [$10..1000]);

%Y Cf. A001358, A001238, A019549. Contains A107342.

%K base,nonn

%O 1,1

%A _Zak Seidov_ and _Robert Israel_, Aug 15 2023