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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 (list; graph; refs; listen; history; text; internal format)
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
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, 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.
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, #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
CROSSREFS
Sequence in context: A047985 A061763 A088474 * A038364 A151360 A329106
KEYWORD
nonn,easy,base
AUTHOR
Marc LeBrun, Oct 20 2003
STATUS
approved

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Last modified June 16 11:31 EDT 2024. Contains 373429 sequences. (Running on oeis4.)