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A306920 a(n) is the smallest prime > 10 where a string of exactly n zeros can be inserted somewhere into the decimal expansion such that the resulting number is also prime.

%I A306920 #27 Jun 02 2021 15:04:35
%S A306920 11,19,17,13,13,23,17,17,31,13,23,41,137,61,23,13,13,67,53,89,19,107,
%T A306920 17,29,61,263,31,37,127,53,269,199,137,23,31,89,61,13,43,163,53,131,
%U A306920 109,19,79,283,109,19,269,223,97,97,223,89,13,79,67,107,17,389,197
%N A306920 a(n) is the smallest prime > 10 where a string of exactly n zeros can be inserted somewhere into the decimal expansion such that the resulting number is also prime.
%C A306920 For many small n, if the decimal expansion of a(n) contains the digit 0, then a(n+1) is a(n) with one zero digit removed. However, this is not true in general. The counterexamples' indices in this sequence are given by A344860.
%H A306920 Felix Fröhlich, <a href="/A306920/b306920.txt">Table of n, a(n) for n = 1..2300</a>
%e A306920 For n = 13: If a string of 13 zeros is inserted between the digits 1 and 3 in 137, the resulting number is 1000000000000037, which is prime. Since 137 is the smallest prime where such a string of 13 zeros can be inserted to get another prime, a(13) = 137.
%o A306920 (PARI) insert(n, len, pos) = my(d=digits(n), v=[], w=[]); for(y=1, pos, v=concat(v, d[y])); v=concat(v, vector(len)); for(z=pos+1, #d, v=concat(v, d[z])); subst(Pol(v), x, 10)
%o A306920 a(n) = forprime(p=10, , for(k=1, #digits(p)-1, my(zins=insert(p, n, k)); if(ispseudoprime(zins), return(p))))
%Y A306920 Column 1 of A306926.
%Y A306920 Cf. A159236, A164329, A215417, A290174, A344860.
%K A306920 nonn,base
%O A306920 1,1
%A A306920 _Felix Fröhlich_, Mar 16 2019