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%I #26 Feb 09 2017 12:04:44
%S 3,5,7,17,25,32,37,40,61,65,85,144,151,162,376,436,645,728,729,908,
%T 1182,1503,1661,2148,2221,2643,3779
%N Numbers n for which the shortest prime right truncation of n^n (in decimal, where a prime exists) sets a record.
%C A220453 contains numbers k not powers of 10 for which no prime at all is formed by truncating rightmost digits of k^k. This sequence includes ties for numbers of digits but not value of the record prime so that 4 is excluded while 5 is not (Leading digits of 3^3, 4^4 and 5^5 are 2, 2 and 3, respectively), and similarly 17 is listed while it would not be if length were the determiner of record -- in fact the primes for 7 and 17 are successor primes 823 and 827. 40 is also such a case.
%C Over the long haul the leading digit string of k^k should encompass all possible values with relative frequencies following Benford's Law, with tendency towards a uniform distribution of klog(k) modulo 1 as k is selected randomly from below some large value. This guarantees the sequence to be infinite by a simple application of the Prime Number Theorem or in other ways. Explicitly also, near certain values the leading string is predictable, so that for instance by finding an integral power of e that has a particular start we can guarantee an infinite sequence of specific values k such that k^k also has this start, namely k=10^M+r for all M>m(r,D) where m(r,D) is some value corresponding to the specific power of e, e^r, and D is the number of digits of it desired.
%C The preceding demonstrates--or argues clearly--that this sequence is infinite. a(27) gives a number of 10058 digits without leading-digit primes. 65^65 has leading digit prime of 65 digits. A question is whether any other value has such a prime as first prime. The sequence A211414 deals with values for which the leading k digits of k^k form a prime.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Benford%27s_law">Benford's Law</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%e The smallest leading prime in the number 7^7 is 823, and then for k=8 through k=15 (excluding k=10) there is a prime smaller than 823 in leading digits of k^k, while 16^16 entirely lacks a prime left segment (and so is listed in A220453). 17^17 begins 827, and this is a new record.
%Y Cf. A220453, A211414.
%K nonn,base
%O 1,1
%A _James G. Merickel_, Dec 15 2012
%E a(27) added by _James G. Merickel_, Feb 12 2013
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