

A087097


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


26



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 n 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 subset of A011539.
Also, numbers n such that the lunar sum of the lunar prime divisors of n is n.  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

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..22095 [all primes with at most 5 digits]
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic


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

Cf. A087019, A087061, A087062, A087636, A087638, A087984.
Sequence in context: A047985 A061763 A088474 * A038364 A151360 A109276
Adjacent sequences: A087094 A087095 A087096 * A087098 A087099 A087100


KEYWORD

nonn,easy,base


AUTHOR

Marc LeBrun, Oct 20 2003


STATUS

approved



