%N For index k = A269230(n), the least prime with k consecutive digits 0, divided by 10^(k+1) and rounded down.
%C For indices k not listed in A269230, the least prime with k digits '0', A037053(k), has these digits consecutively, in a single run. If k is listed in A269230, this is not the case (e.g., A037053(32) = 10...0603), and the most economical way to make a prime with k consecutive digits 0 is to put two (a priori nonzero) digits in front of the string of k '0's, i.e., p = a*10^(k+1) + b with a > 9.
%C This sequence lists these numbers a, and the corresponding prime (least prime with k consecutive digits 0) is simply nextprime(a*10^(k+1)).
%C If a is a multiple of 10, then b can have two nonzero digits, 11 <= b <= 99. Otherwise (b < 10), this prime is also the least prime with k+1 (consecutive) digits '0', A037053(k+1), and k+1 is listed in A085824 (unless a > 90). It is then obviously not the smallest prime with *exactly* k consecutive digits 0, but with *at least* k consecutive digits 0. This happens for (n,k,a,b) = (2,43,10,9), (24,108,10,7), (28,121,10,3), (34,132,10,7), (38,144,10,9), ...
%o (PARI) A269260(n,k=A269230(n))=for(a=1,9e9,nextprime(a*10^(k+1))-a*10^(k+1)<10^(valuation(a,10)+1)&&return(a)) \\ If the 2nd (optional) arg is given, the 1st arg 'n' is ignored. Otherwise the function A269230() must be defined.
%A _M. F. Hasler_, Feb 22 2016