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A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

%I A087416 #29 Jan 24 2022 08:01:47
%S A087416 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 A087416 29,29,29,99,99,99,99,39,39,39,39,39,39,99,99,99,99,99,49,49,49,49,49,
%U A087416 99,99,99,99,99,99,59,59,59,59,99,99,99,99,99,99,99,69,69,69
%N A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.
%C A087416 Two comments from _David Applegate_ on lunar perfect numbers, Nov 08 2003: (Start)
%C A087416 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 A087416 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 A087416 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 A087416 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 A087416 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 A087416 <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>
%K A087416 nonn,easy,base
%O A087416 1,1
%A A087416 Marc LeBrun and _N. J. A. Sloane_, Oct 19 2003
%E A087416 More terms from _David Applegate_, Nov 07 2003