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.

2, 73, 373, 3733, 23333, 719333, 2339933, 23399339

Offset: 1

Author

James G. Merickel, Apr 02 2014

Keywords

• nonn
• fini
• base

Comments

When the numbers here are concatenated through to the end, the factorizations are:

3 * 7 * 13,

461 * 593,

83 * 32936551,

151 * 1810419690883,

(3^3) * 827 * 12242974308000077,

7 * 73 * 173 * 727 * 4409 * 9647498037197777, and

100003 * 2733651723681626744530004033113.

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)).

The exponents producing these leading digits are 1, 2, 75, 75, 1562, 17573, 454667, and 3471613.

Examples

			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).
		

Links

• Wikipedia, Truncatable prime
• Wikipedia, Mathematical coincidence
• James G. Merickel, curio on powers of e (1st appearing leading right-truncatables of each length concatenated)
• James G. Merickel, curio on powers of e (1st appearing leading right-truncatables of each length concatenated), message 25448 in primenumbers Yahoo group, Jan 7, 2014.

Crossrefs

Cf. A024770.

Programs

PARI

{
\\ R is the array of 8 by-length ordered lists of right-truncatable primes.\\
\\ a is the vector of list-sizes for R.\\
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]];
a=[4,9,14,16,15,12,8,5];i=1;e=exp(1);e1=e/10;n=e;
for(j=1,8,
  E=10^j;while(1,
    m=floor(n);for(k=1,a[j],
      if(m==R[j][k],print(m);n*=10;break(2)));
    if(n>E/e,n*=e1,n*=e);i++))
}