

A087097


Lunar primes (formerly called dismal primes) (cf. A087062).


28



19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 209, 219, 309, 319, 329, 409, 419, 429, 439, 509, 519, 529, 539, 549, 609, 619, 629, 639, 649, 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
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OFFSET

1,1


COMMENTS

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.
All lunar primes must contain a 9, so this is a subsequence of A011539.
Also, numbers k such that the lunar sum of the lunar prime divisors of k is k.  N. J. A. Sloane, Aug 23 2010
We have changed the name from "dismal arithmetic" to "lunar arithmetic"  the old name was too depressing.  N. J. A. Sloane, Aug 06 2014
(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


LINKS

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.


FORMULA

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


EXAMPLE

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.


PROG

(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, #d1, forvec(v=vector(L, i, [i==1, 9]), vecmax(n)<9&&next; m=fromdigits(v); for(k=10^(#dL), 10^(#dL+1)1, A087062(m, k)==n&&return))))}, [1..999])) \\ M. F. Hasler, Nov 16 2018


CROSSREFS



KEYWORD

nonn,easy,base


AUTHOR



STATUS

approved



