First n-digit prime in the digits of sqrt(2).
2, 41, 421, 5623, 56237, 135623, 3562373, 42135623, 213562373, 4142135623, 85696718753, 969807856967, 6948073176679, 67973799073247, 980785696718753, 4880168872420969, 78462107038850387, 737990732478462107, 1875376948073176679, 69480731766797379907
1
The position of these numbers in given in S000711. See S000712 and S000714 for the primes in sqrt(3) and sqrt(Pi).
T. D. Noe, Plot of 300 terms
T. D. Noe, Table of 300 terms
Wikipedia, Square root of 2
(Mma) d = RealDigits[Sqrt[2], 10, 1000]; t = {}; n = 1; While[d1 = Partition[d[[1]], n, 1]; s = Select[d1, Ceiling[Log[10, FromDigits[#]]] == n && PrimeQ[FromDigits[#]] &, 1]; s != {}, AppendTo[t, {FromDigits[s[[1]]], Position[d1, s[[1]], 1, 1][[1, 1]]}]; n++]; Transpose[t][[1]]
Cf. A002193 (digits of sqrt(2)), S000711, S000712, S000714.
nonn,base
T. D. Noe, Oct 06 2015