%I
%S 1319732172923115,1713323729219115,2191317297323115,2291117192373135,
%T 2291772319133115,2322931117719135,2329131171719235,7112917313223195,
%U 7191317229231135,7191331129217235
%N Concatenations of the first k primes for some k that are also products of k distinct primes.
%C All but one of the above are unique concatenations of the first 10 primes, with the 7th member being representable as 2 different such concatenations. No concatenation of (more than 1 and) fewer than 10 initial primes is also a product of the same number of distinct primes, and a search of concatenations of the first 11, 12 and 13 primes revealed no further members of this sequence. Though it is undetermined that there are no more, the sequence should certainly be finite. A concatenation of any number of the first primes is small enough to be a product of the same number of distinct primes, but a coincidence for a large number would be statistically unlikely as the intersection of two sets of numbers small in numbers compared to their sizes. This sequence's discovery dates to around a year and a half before this submission.
%C Base 10 hits a 'sweet spot' for this question: For all other bases from 2 to 600000, there is a concatenation of some number of primes less than 7 that is a product of that many distinct primes.
%e In terms of an alphabetical ordering in terms of factorization, the largest element here is first, as 3*5*7*11*13*19*37*167*821*4969; and the next largest is next in line, as 3*5*7*11*13*37*71*107*577*2953.
%K nonn,base,fini
%O 1,1
%A _James G. Merickel_, Oct 10 2012
%E Search limit extended through concatenations of the first 13 primes by _James G. Merickel_, Aug 30 2013
