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Considering proper Let f(M,k) denote the decimal concatenations f(M,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.

The definition is not clear to me. - N. J. A. Sloane, Aug 11 2015

nonn,base,more,hard,changed,obsc

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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 in this case 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)).

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Considering proper decimal concatenations f(M,k) = M | M-1 | M-2 | ... | M-k+1, k > 1, 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.

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Considering proper decimal concatenations f(M,k) = M | M-1 | M-2 | ... | M-k+1, k > 1, 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(m), , order prioritized.

The alternative sequence where only the rapidity of arrival of the n-th prime determines a(n) (the title's k(n) minimal, but for the largest prime, with no constraint on k(m), m < n 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 in this case 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)).

Considering proper decimal concatenations f(M,k) = M | M-1 | M-2 | ... | M-k+1, k > 1, 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(m), order prioritized m = 1 through n.