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A135377 Smallest n-primeval prime, i.e., minimal prime number containing all A006880(n) primes < 10^n embedded in it as permutations of some of its substrings.

%I A135377 #7 Oct 13 2013 10:03:49
%S A135377 2357,1123465789,10112233445566788997,100111222333444555666777998889,
%T A135377 1000111222233334444555666777798889899,
%U A135377 100001111222233333444445555566666777778888999989
%N A135377 Smallest n-primeval prime, i.e., minimal prime number containing all A006880(n) primes < 10^n embedded in it as permutations of some of its substrings.
%C A135377 a(1) - a(4) were computed by Mike Keith in 2008 and a(4) - a(31) by _Jérôme STORTI_ in 2002.
%H A135377 M. Keith, <a href="http://www.cadaeic.net/primeval.htm">Integers containing many embedded primes</a>
%e A135377 Mike Keith's website uses a shorthand notation for these numbers. The 4-primeval prime 100111222333444555666777998889 is written in this notation as (1) 2 3 3 3 3 3 3 3 0 998889. The (1) represents the leading 1 digit (which will always be present). The next number says how many consecutive 0's follow the leading 1 and the next says how many consecutive 1's follow that and so on up to the number of consecutive 8's. The final grouping explicitly shows how the last group of 8's and 9's are arranged.
%e A135377 The following are the n-primeval primes as found by _Jérôme STORTI_ in this notation:
%e A135377 5 (1) 3 3 4 4 4 3 3 4 0 98889899
%e A135377 6 (1) 4 4 4 5 5 5 5 5 4 999989
%e A135377 7 (1) 5 5 5 6 5 5 5 6 3 98899999
%e A135377 8 (1) 5 6 7 7 6 7 7 7 6 98999999
%e A135377 9 (1) 7 7 8 8 8 7 8 8 6 9999989899
%e A135377 10 (1) 8 8 8 9 9 9 9 9 7 9999899999
%e A135377 11 (1) 8 9 10 10 10 9 10 10 6 9889989999999
%e A135377 12 (1) 10 10 10 11 11 11 10 11 9 9998999999899
%e A135377 13 (1) 10 11 11 12 11 12 11 12 9 99899999999899
%e A135377 14 (1) 11 13 13 13 12 12 12 13 11 989999989999999
%e A135377 15 (1) 12 13 14 14 13 14 13 14 12 9999999988999999
%e A135377 16 (1) 13 14 14 15 14 14 14 15 12 99999999999999889
%e A135377 17 (1) 14 15 15 16 15 15 15 16 14 998999999999998999
%e A135377 18 (1) 16 17 17 17 16 17 17 17 14 9989999999999899999
%e A135377 19 (1) 17 18 17 18 17 17 17 18 15 988999999899999999999
%e A135377 20 (1) 17 19 18 19 19 18 19 19 16 999999998999999999989
%e A135377 21 (1) 18 19 19 20 19 19 20 20 17 9899999999999999998999
%e A135377 22 (1) 18 20 20 21 20 21 21 21 18 99998999999999999998999
%e A135377 23 (1) 21 23 21 22 21 21 22 22 19 999999889999999999999999
%e A135377 24 (1) 20 22 22 23 22 22 22 23 21 999999999999999989999999
%e A135377 25 (1) 23 23 23 24 23 23 23 24 22 9999999999999999998999999
%e A135377 26 (1) 23 24 24 25 25 25 24 25 22 999999999999999999899999989
%e A135377 27 (1) 24 25 25 26 25 25 25 26 23 9999999998999999999999998999
%e A135377 28 (1) 25 26 27 27 27 26 27 27 25 9999899999999999999999999999
%e A135377 29 (1) 25 27 27 28 27 27 27 28 25 999999989999999999999999999989
%e A135377 30 (1) 26 29 28 29 29 28 28 29 27 999999999999998999999999999999
%e A135377 31 (1) 28 29 29 30 29 29 29 30 27 99999889999999999999999999999999
%e A135377 a(2) = 1123465789 because this is the smallest prime out of which each of the first 25 primes below 10^2, viz. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 can be formed using its digits.
%Y A135377 Cf. A072857, A039993, A134649.
%K A135377 nonn,base
%O A135377 1,1
%A A135377 _Lekraj Beedassy_, Dec 09 2007
%E A135377 Link fixed by _Charles R Greathouse IV_, Aug 13 2009