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%I #98 Aug 11 2015 13:36:59
%S 4,10,1000,21910420,1113475000,67483920430
%N Let f(M,k) denote the decimal concatenation of k numbers starting with M: M | M-1 | M-2 | ... | M-k+1, k > 1. Then a(n) is the smallest M such that for all m in {1,..,n} an m-th prime occurs as f(M,k) for the smallest possible k, order prioritized m = 1 through n.
%C The sequence of k values is 2, 4, 8, 10, 14, 20, 28, 32, 34, 40, 50, etc. a(n) is necessarily congruent to 10 modulo 30 starting with n=4 (and is coincidentally so for n=2 and 3). Each successive unknown term's existence is only conjectural, but is supported by standard heuristics. The values in the lead sentence come through considerations modulo the smallest primes.
%C Concatenation of 22 numbers is ruled out by consideration modulo 11 in concert with the prior need for the stem to be 10 mod 30: A prime concatenation of 22 values necessarily involves passage from one length number to another, with a power of 10 not leading. Beginning with a(11), however, with each time a multiple of 22 for the number of concatenated values is passed there is a measure of uncertainty. Theoretically, it seems a(11) could include for largest prime a concatenation of 44 (rather than 50) values, but with these primes very sparse it is a near certainty heuristically that this is not so. Mathematically, a conjectured a(11) would have a higher bar for strict proof than a(n) for n < 11, and the same holds repeatedly for even less accessible terms.
%C The alternative sequence where only the rapidity of arrival of the n-th prime determines a(n) (k minimal for the largest prime, with no constraint on k for the smaller prime concatenations) necessarily shares its first 5 terms in common with this one. It also shares its 6th by virtue of the fact that this sequence's a(6) is the only value less than 10^12 producing its 6th prime with the attachment of the 20th value, whether alternative length possibilities for primes are allowed or not (i.e., the first cases giving 5 other smaller primes -- in addition to one of 20 concatenated values -- where there is a prime concatenation of 16 values, in place of one of either 2 or 8 values, are both at least this large). However, it does necessarily differ at a(7) and a(8) (but then not necessarily at a(9)), as the resolution of the theoretical problem for the twin sequence is given for a(7) by the possibility of 5, 7, 11, 13, 17, 23 and 25 numbers being concatenated to give primes, and for a(8) by the replacement of 2 concatenated values with concatenations of both 16 and 26 of them (with result that a(8) for this alternative sequence appears already with concatenation of 28 values, while here that corresponds to a(7)).
%C This necessity to distinguish between whether or not only the last of the primes comes as quickly as possible does not arise as an issue in the situation where the number itself is required to be prime (A172257). [Comments re-edited from Feb 2014] - _James G. Merickel_, Aug 07 2015
%C The definition is not clear to me. - _N. J. A. Sloane_, Aug 11 2015
%e 43 is prime while 32 and 21 are not, so a(1)=4; 109 and 10987 are both prime, and like concatenations for values 4 through 9 do not produce 2 primes, so a(2)=10; 1000999, 1000999998997 and 1000999998997996995994993 are all prime and no smaller value produces 3 primes so quickly, so a(3)=1000.
%Y Cf. A152397, A172257, A232657.
%K nonn,base,more,hard,obsc
%O 1,1
%A _James G. Merickel_, Oct 20 2009
%E Two more terms from _James G. Merickel_, Dec 09 2009
%E 6th term added by _James G. Merickel_, Jan 29 2010
%E Title changed by _James G. Merickel_, Feb 18 2014
%E Title changed by _James G. Merickel_, Aug 06 2015
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