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.

11, 19, 17, 13, 13, 23, 17, 17, 31, 13, 23, 41, 137, 61, 23, 13, 13, 67, 53, 89, 19, 107, 17, 29, 61, 263, 31, 37, 127, 53, 269, 199, 137, 23, 31, 89, 61, 13, 43, 163, 53, 131, 109, 19, 79, 283, 109, 19, 269, 223, 97, 97, 223, 89, 13, 79, 67, 107, 17, 389, 197

Offset: 1

Author

Felix Fröhlich, Mar 16 2019

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..2300

Crossrefs

Column 1 of A306926.

Cf. A159236, A164329, A215417, A290174, A344860.

Programs

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)
a(n) = forprime(p=10, , for(k=1, #digits(p)-1, my(zins=insert(p, n, k)); if(ispseudoprime(zins), return(p))))