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A240174 a(n) is the right-truncatable prime of n digits appearing as the initial digits of the smallest number of the form exp(k) for some positive integer k.

%I A240174 #28 Nov 13 2019 15:21:14
%S A240174 2,73,373,3733,23333,719333,2339933,23399339
%N A240174 a(n) is the right-truncatable prime of n digits appearing as the initial digits of the smallest number of the form exp(k) for some positive integer k.
%C A240174 When the numbers here are concatenated through to the end, the factorizations are:
%C A240174   3 * 7 * 13,
%C A240174   461 * 593,
%C A240174   83 * 32936551,
%C A240174   151 * 1810419690883,
%C A240174   (3^3) * 827 * 12242974308000077,
%C A240174   7 * 73 * 173 * 727 * 4409 * 9647498037197777, and
%C A240174   100003 * 2733651723681626744530004033113.
%C A240174 As the probability of a random string of 4 digits all matching (selected uniformly over {0,1,2,3,...,9}) is one in 1/1000, the coincidence here by a criterion involving this feature would make these concatenations' factorizations seem somewhat freakish.  There appears to be little guidance in the mathematical literature in English on measuring or rank-ordering what appear to be digital or other types of coincidence (see, for example, the reference on this subject as it currently stands at Wikipedia (below)).
%C A240174 The exponents producing these leading digits are 1, 2, 75, 75, 1562, 17573, 454667, and 3471613.
%H A240174 Wikipedia, <a href="http://en.wikipedia.org/wiki/Truncatable_prime">Truncatable prime</a>
%H A240174 Wikipedia, <a href="http://en.wikipedia.org/wiki/Mathematical_coincidence">Mathematical coincidence</a>
%H A240174 James G. Merickel, <a href="http://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/25448">curio on powers of e (1st appearing leading right-truncatables of each length concatenated)</a>
%H A240174 James G. Merickel, <a href="/A240174/a240174.txt">curio on powers of e (1st appearing leading right-truncatables of each length concatenated)</a>, message 25448 in primenumbers Yahoo group, Jan 7, 2014.
%e A240174 2 is the leading single digit of e itself and is by the convention of A024770 considered truncatable; the leading digits of e^2, without decimal, are the right-truncatable 73; and e^75 is then the first to produce a 3-digit right-truncatable prime, also producing the 4-digit one (a(3)=373 and a(4)=3733, with e^75 beginning with these digits).
%o A240174 (PARI)
%o A240174 {
%o A240174 \\ R is the array of 8 by-length ordered lists of right-truncatable primes.\\
%o A240174 \\ a is the vector of list-sizes for R.\\
%o A240174 R=[[2,3,5,7],[23,29,31,37,53,59,71,73,79],[223,239,293,311,313,317,373,379,593,599,719,733,739,797],[2333,2339,2393,2399,2939,3119,3137,3733,3739,3793,3797,5939,7193,7331,7333,7393],[23333,23339,23399,23993,29399,31193,31379,37337,37339,37397,59393,59399,71933,73331,73939],[233993,239933,293999,373379,373393,593933,593993,719333,739393,739397,739399],[2339933,2399333,2939999,3733799,5939333,7393913,7393933],[23399339,29399999,37337999,59393339,73939133]];
%o A240174 a=[4,9,14,16,15,12,8,5];i=1;e=exp(1);e1=e/10;n=e;
%o A240174 for(j=1,8,
%o A240174   E=10^j;while(1,
%o A240174     m=floor(n);for(k=1,a[j],
%o A240174       if(m==R[j][k],print(m);n*=10;break(2)));
%o A240174     if(n>E/e,n*=e1,n*=e);i++))
%o A240174 }
%Y A240174 Cf. A024770.
%K A240174 nonn,fini,base
%O A240174 1,1
%A A240174 _James G. Merickel_, Apr 02 2014