A365009 Semiprimes that are the concatenation of two or more semiprimes.

46, 49, 69, 94, 106, 146, 159, 214, 219, 226, 254, 259, 334, 339, 346, 386, 394, 415, 422, 446, 451, 458, 466, 469, 482, 485, 493, 514, 519, 554, 559, 579, 586, 589, 614, 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

Offset: 1

Author

Zak Seidov and Robert Israel, Aug 15 2023

Keywords

• base
• nonn

Comments

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?

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A001238, A019549. Contains A107342.

Programs

Maple

filter:= proc(n) local d,v;
  if numtheory:-bigomega(n) <> 2 then return false fi;
  for d from 1 to length(n)-1 do
     v:= n  mod 10^d;
     if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and g((n-v)/10^d) then return true fi
  od;
  false
end proc:
g:= proc(n) local d,v; option remember;
  if numtheory:-bigomega(n) = 2 then return true fi;
  for d from 1 to length(n)-1 do
    v:= n mod 10^d;
    if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and procname((n-v)/10^d) then return true fi
  od;
  false
end proc:
select(filter, [$10..1000]);