%I #31 Jan 06 2018 18:46:20

%S 29,599,7,2999,97,47,179999,1999,1097,89,23999999,69997,21997,1193,

%T 139,23999999,199999,369997,23993,691,199,29999999,19999999,3199997,

%U 149993,10993,199,113,17399999999,19999999,6999997,1199999,139999,997,293,1831

%N Square array read by antidiagonals upwards: M(n,k) is the initial occurrence of first prime p1 of consecutive primes p1, p2, where p2 - p1 = 2*k, and p1, p2 span a multiple of 10^n, n>=1, k>=1.

%C The unit digits of the numbers in the matrix representation M(n,k) are 9's for column 1, 7's or 9's for column 2, 7's for column 3, 3's or 9's for column 4, and 1's, 3's, 7's or 9's for column 5.

%C The following matrix terms appear as first terms in sequence

%C A060229(1) = M(1,1)

%C A288021(1) = M(1,2)

%C A288022(1) = M(1,3)

%C A288024(1) = M(1,4)

%C A031928(1) = M(1,5)

%C A158277(1) = M(2,1)

%C A160440(1) = M(2,2)

%C A160370(1) = M(2,3)

%C A287049(1) = M(2,4)

%C A160500(1) = M(2,5)

%C A158861(1) = M(3,1).

%F M(n,k) = min( p_i : p_(i+1) - p_i = 2*k, p_i and p_(i+1) consecutive primes and p_i < m*10^n < p_(i+1) for some integer m) where p_j is the j-th prime, n>=1 and k>=1.

%e The matrix representation of the sequence with row n indicating the spanned power of 10 and column k indicating the difference of 2*k between the first pair of consecutive primes spanning a multiple of 10^n:

%e --------------------------------------------------------------------------

%e n\k 1 2 3 4 5

%e --------------------------------------------------------------------------

%e 1 | 29 7 47 89 139

%e 2 | 599 97 1097 1193 691

%e 3 | 2999 1999 21997 23993 10993

%e 4 | 179999 69997 369997 149993 139999

%e 5 | 23999999 199999 3199997 1199999 1999993

%e 6 | 23999999 19999999 6999997 38999993 1999993

%e 7 | 29999999 19999999 159999997 659999999 379999999

%e 8 | 17399999999 7699999999 9399999997 8999999993 499999993

%e 9 | 92999999999 135999999997 85999999997 8999999993 28999999999

%e 10| 569999999999 519999999997 369999999997 29999999993 819999999997

%e ...

%e Every column in the matrix is nondecreasing.

%e For the first and fourth columns, ceiling(M[n,1]/10^n) and ceiling(M[n,4]/10^n) are divisible by 3, for all n>=1 (see A158277 and A287049).

%Y Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049.

%Y Cf. A060229, A288021, A288022, A288024.

%K nonn,tabl

%O 1,1

%A _Hartmut F. W. Hoft_, May 18 2017