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Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.
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%I #29 Jan 24 2022 08:01:47

%S 9,9,9,9,9,9,9,9,9,99,99,19,19,19,19,19,19,19,19,99,99,99,29,29,29,29,

%T 29,29,29,99,99,99,99,39,39,39,39,39,39,99,99,99,99,99,49,49,49,49,49,

%U 99,99,99,99,99,99,59,59,59,59,99,99,99,99,99,99,99,69,69,69

%N Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

%C Two comments from _David Applegate_ on lunar perfect numbers, Nov 08 2003: (Start)

%C 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.

%C 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)

%H D. Applegate, <a href="/A087061/a087061.txt">C program for lunar arithmetic and number theory</a> [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="http://arxiv.org/abs/1107.1130">Dismal Arithmetic</a> [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Sloane/carry2.html">Dismal Arithmetic</a>, J. Int. Seq. 14 (2011) # 11.9.8.

%H <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>

%K nonn,easy,base

%O 1,1

%A Marc LeBrun and _N. J. A. Sloane_, Oct 19 2003

%E More terms from _David Applegate_, Nov 07 2003