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A287050 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.

%I A287050 #31 Jan 06 2018 18:46:20
%S A287050 29,599,7,2999,97,47,179999,1999,1097,89,23999999,69997,21997,1193,
%T A287050 139,23999999,199999,369997,23993,691,199,29999999,19999999,3199997,
%U A287050 149993,10993,199,113,17399999999,19999999,6999997,1199999,139999,997,293,1831
%N A287050 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 A287050 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 A287050 The following matrix terms appear as first terms in sequence
%C A287050 A060229(1) = M(1,1)
%C A287050 A288021(1) = M(1,2)
%C A287050 A288022(1) = M(1,3)
%C A287050 A288024(1) = M(1,4)
%C A287050 A031928(1) = M(1,5)
%C A287050 A158277(1) = M(2,1)
%C A287050 A160440(1) = M(2,2)
%C A287050 A160370(1) = M(2,3)
%C A287050 A287049(1) = M(2,4)
%C A287050 A160500(1) = M(2,5)
%C A287050 A158861(1) = M(3,1).
%F A287050 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 A287050 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 A287050 --------------------------------------------------------------------------
%e A287050 n\k   1             2             3             4            5
%e A287050 --------------------------------------------------------------------------
%e A287050 1 |   29            7             47            89           139
%e A287050 2 |   599           97            1097          1193         691
%e A287050 3 |   2999          1999          21997         23993        10993
%e A287050 4 |   179999        69997         369997        149993       139999
%e A287050 5 |   23999999      199999        3199997       1199999      1999993
%e A287050 6 |   23999999      19999999      6999997       38999993     1999993
%e A287050 7 |   29999999      19999999      159999997     659999999    379999999
%e A287050 8 |   17399999999   7699999999    9399999997    8999999993   499999993
%e A287050 9 |   92999999999   135999999997  85999999997   8999999993   28999999999
%e A287050 10|   569999999999  519999999997  369999999997  29999999993  819999999997
%e A287050 ...
%e A287050 Every column in the matrix is nondecreasing.
%e A287050 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 A287050 Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049.
%Y A287050 Cf. A060229, A288021, A288022, A288024.
%K A287050 nonn,tabl
%O A287050 1,1
%A A287050 _Hartmut F. W. Hoft_, May 18 2017