A269260 For index k = A269230(n), the least prime with k consecutive digits 0, divided by 10^(k+1) and rounded down.

19, 10, 16, 16, 20, 30, 20, 15, 30, 26, 23, 27, 19, 17, 40, 30, 13, 13, 13, 24, 28, 22, 20, 10, 20, 30, 16, 10, 40, 13, 16, 11, 39, 10, 20, 20, 30, 10, 23, 16, 15, 30, 34, 56, 19, 28, 20, 20, 30, 20, 20, 90, 87, 68, 20, 25, 20, 16, 30, 40

Offset: 1

Author

M. F. Hasler, Feb 22 2016

Keywords

• nonn
• base

Comments

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.

This sequence lists these numbers a, and the corresponding prime (least prime with k consecutive digits 0) is simply nextprime(a*10^(k+1)).

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), ...

Programs

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.