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A083427 Smallest prime which is a concatenation of n distinct primes.

%I A083427 #27 Mar 04 2020 16:55:20
%S A083427 2,23,257,2357,112573,11132357,1113223537,111317193257,11131719223357,
%T A083427 1113171922335437,111317192232934157,11131719223293135773,
%U A083427 1113171922329313375759,111317192232931337415743,11131719223293133741435717
%N A083427 Smallest prime which is a concatenation of n distinct primes.
%e A083427 a(5) = 112573 is a concatenation of 11,2,5,7 and 3 and is the smallest such prime.
%e A083427 a(7) <= 1113223537 = 11//13//2//23//5//3//7. - _R. J. Mathar_, Mar 19 2011
%e A083427 a(8) <= 111317193257 = 11//13//17//19//3//2//5//7. - _Jonathan Vos Post_, Mar 20 2006
%e A083427 a(9) <= 11131719223357 = 11//13//17//19//2//23//3//5//7. - _R. J. Mathar_, Mar 19 2011
%o A083427 (Sage)
%o A083427 concat = lambda x: Integer(''.join(str(i) for i in x),base=10)
%o A083427 def A083427(n):
%o A083427     def primelists(sofar, widths):
%o A083427         if not widths: yield sofar; return
%o A083427         w = widths[0]
%o A083427         for p in prime_range(10**(w-1), 10**w):
%o A083427             if p not in sofar:
%o A083427                 for pv in primelists(sofar+[p], widths[1:]):
%o A083427                     yield pv
%o A083427     for numdig in PositiveIntegers():
%o A083427         least = None
%o A083427         for part in Partitions(numdig, length=n):
%o A083427             if list(part).count(1) > 4: continue # optimization
%o A083427             for sizes in Permutations(part):
%o A083427                 for plist in primelists([], sizes):
%o A083427                     x = concat(plist)
%o A083427                     if is_prime(x): least = min(x, least) if least else x
%o A083427                     # since x is increasing in this inner loop,
%o A083427                     # no need to continue if we can't improve
%o A083427                     if least and x >= least: break
%o A083427         if least: return least # _D. S. McNeil_, Mar 20 2011
%Y A083427 Cf. A000040.
%K A083427 base,more,nonn
%O A083427 1,1
%A A083427 _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 29 2003
%E A083427 a(7), a(8) from _Jonathan Vos Post_, Mar 20 2006
%E A083427 a(7) corrected by _Emmanuel Vantieghem_, Mar 19 2011
%E A083427 a(8) deleted on grounds that it is quite likely to be wrong. - _N. J. A. Sloane_, Mar 19 2011
%E A083427 a(7)-a(15) from _D. S. McNeil_, Mar 20 2011