Certain entries require n+3 digits such as a(13). Those which do not require three digits besides n zeros are in
A085824. Conjecture: a prime of this form,
A037053, requires at most three judiciously placed nonzero digits, two on the exterior and one inside. -
Robert G. Wilson vThe zeros in a(n) do not need to be consecutive. If the zeros must be consecutive we get a new sequence which agrees with this one up though a(31), but then here we have a(32) = 10000000000000000000000000000000603, whereas the smallest prime with exactly 32 consecutive zeros is 19000000000000000000000000000000009 = 19*10^33+9. -
N. J. A. Sloane, Feb 20 2016
Sequence
A269230 lists the indices (32, 43, 46, 49, 50, 60, 69, ...) for which a(n) does not have n consecutive digits '0', and gives more information about the smallest prime which has n consecutive digits '0': Sequence
A269260 lists the values a > 9 such that the least prime with n *consecutive* '0's equals nextprime(a*10^(n+1)), for these indices n. -
M. F. Hasler, Feb 20 2016 and Feb 22 2016
Since the definition requires "exactly" and not "at least" n 0's, the sequence is not increasing, e.g., a(22) = 10^24 + 49 > a(23) = 10^24 + 7. However, it seems that no term has more digits than its successor. -
M. F. Hasler, Feb 20 2016
Conjecture 1: (following
Robert G. Wilson v's conjecture above and insight provided by Hans Havermann): terms with two substrings of zeros have first digit of 1.
Conjecture 2: as n increases, there are more terms with two substrings of zeros than one consecutive string.
The logic is as follows: Let n = number of zeros and z = number of substrings of consecutive zeros contained in candidate primes. Candidates with two substrings of zeros (z=2) must be considered after only 72 smaller z=1 candidates have been considered and excluded, i.e., numbers of the form a[n zeros]b and 1[n zeros]ab, where a in {1..9}, b in {1,3,7,9}. After these 72 candidates are excluded, 36*n z=2 candidates are considered before having to consider only 36 additional z=1 candidates (i.e., 2[n zeros]ab), followed by 36*n additional z=2 candidates, etc. So as n increases, it becomes increasingly unlikely that any z=1 term appears. Additionally, the number of candidates increases as n increases for z>=3. For a given n, 1044 + 324*(n+1) candidates must be excluded before considering the smallest z=3: 1[n-2 zeros]10101. Since the probability p of n-digit primes occurring is p ~ 1/(n log 10) by the Prime Number Theorem, and the number of z in {1,2} candidates which must be excluded increases exponentially as n and z increase, it is unlikely that any z >= 3 term appears in
A037053 and that first digit will be > 1 for any z=2 term.
Sequence
A269233 lists the number of candidate primes < a(n); i.e., the number of excluded candidates.
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