OFFSET
1,1
COMMENTS
Two comments from David Applegate on lunar perfect numbers, Nov 08 2003: (Start)
If we define a perfect number by "n is lunarly perfect if Sum_{d|n} d == 2*n (both sum and * lunar)", no such numbers exist because 9|n, so the lunar sum of divisors ends in 9, but 2*n ends in 2.
If we define a perfect number by "n is lunarly perfect if lunar Sum_{d|n, d != n} d == n", no such numbers exist. For suppose n is perfect. n != 9 (since 9 is 9's only divisor). Then 9|n and 9 != n, so Sum_{d|n, d!=n} d ends in 9 and thus so does n. But 9ish numbers are not divisible by any single digit < 9. Thus n has no divisors of the same length as n, other than n itself. So Sum_{d|n, d!=n} d is one digit shorter than n. (End)
LINKS
D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Marc LeBrun and N. J. A. Sloane, Oct 19 2003
EXTENSIONS
More terms from David Applegate, Nov 07 2003
STATUS
approved