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%I #57 May 08 2021 22:59:01
%S 19,29,39,49,59,69,79,89,90,91,92,93,94,95,96,97,98,99,109,209,219,
%T 309,319,329,409,419,429,439,509,519,529,539,549,609,619,629,639,649,
%U 659,709,719,729,739,749,759,769,809,819,829,839,849,859,869,879,901,902,903,904,905,906,907,908,909,912,913,914,915,916,917,918,919,923,924,925,926,927,928,929,934,935,936,937,938,939,945,946,947,948,949,956,957,958,959,967,968,969,978,979,989
%N Lunar primes (formerly called dismal primes) (cf. A087062).
%C 9 is the multiplicative unit. A number is a lunar prime if it is not a lunar product (see A087062 for definition) r*s where neither r nor s is 9.
%C All lunar primes must contain a 9, so this is a subsequence of A011539.
%C Also, numbers k such that the lunar sum of the lunar prime divisors of k is k. - _N. J. A. Sloane_, Aug 23 2010
%C We have changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing. - _N. J. A. Sloane_, Aug 06 2014
%C (Lunar) composite numbers are not necessarily a product of primes. (For example 1 = 1*x for any x in {1, ..., 9} is not a prime but can't be written as the product of primes.) Therefore, to establish primality, it is not sufficient to consider only products of primes; one has to consider possible products of composite numbers as well. - _M. F. Hasler_, Nov 16 2018
%H David Applegate and N. J. A. Sloane, <a href="/A087097/b087097.txt">Table of n, a(n) for n = 1..22095</a> [all primes with at most 5 digits]
%H D. Applegate, <a href="/A087061/a087061.txt">C program for lunar arithmetic and number theory</a>
%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="http://arxiv.org/abs/1107.1130">Dismal Arithmetic</a>, arXiv:1107.1130 [math.NT], 2011.
%H Brady Haran and N. J. A. Sloane, <a href="https://youtu.be/cZkGeR9CWbk">Primes on the Moon (Lunar Arithmetic)</a>, Numberphile video, Nov 2018.
%H <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>
%F The set { m in A011539 | 9<m<100 or A054054(m) < min(A000030(m),A010879(m)) } (9ish numbers A011539 with 2 digits or such that the smallest digit is strictly smaller than the first and the last digit) is equal to this sequence up to a(1656) = 10099. The next larger 9ish number 10109 is also in that set but is the lunar square of 109, thus not in this sequence of primes. - _M. F. Hasler_, Nov 16 2018
%e 8 is not prime since 8 = 8*8. 9 is not prime since it is the multiplicative unit. 10 is not prime since 10 = 10*8. Thus 19 is the smallest prime.
%o (PARI) A87097=select( is_A087097(n)={my(d); if( n<100, n>88||(n%10==9&&n>9), vecmax(d=digits(n))<9, 0, #d<5, vecmin(d)<min(d[1],d[#d]), my(m); !for(L=#d\/2,#d-1, forvec(v=vector(L,i,[i==1,9]),vecmax(n)<9&&next; m=fromdigits(v); for(k=10^(#d-L),10^(#d-L+1)-1, A087062(m,k)==n&&return))))}, [1..999])) \\ _M. F. Hasler_, Nov 16 2018
%Y Cf. A087019, A087061, A087062, A087636, A087638, A087984.
%K nonn,easy,base
%O 1,1
%A _Marc LeBrun_, Oct 20 2003
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