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A117346 Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

%I A117346 #46 Feb 16 2025 08:33:00
%S A117346 1,3,4,5,6,7,8,10,11,13,16,17,19,20,23,28,29,31,32,37,41,43,47,53,59,
%T A117346 61,64,67,70,71,73,79,83,88,89,97,101,103,104,107,109,110,113,120,127,
%U A117346 128,131,136,137,139,149,151,152,157,163,167,173,179,181,191,193,197,199
%N A117346 Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).
%C A117346 Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.
%D A117346 R. K. Guy, Unsolved Problems in Number Theory, B2.
%H A117346 Amiram Eldar, <a href="/A117346/b117346.txt">Table of n, a(n) for n = 1..10000</a>
%H A117346 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiperfectNumber.html">Multiperfect Number</a>.
%e A117346 70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.
%t A117346 asmlQ[n_]:=Module[{p=Mod[DivisorSigma[1,n],n]},If[p>n/2,p=n-p];p<=Log[n]];
%t A117346 Select[Range[200],asmlQ] (* _Harvey P. Dale_, Dec 25 2013 *)
%Y A117346 Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117347 through A117350.
%K A117346 nonn
%O A117346 1,2
%A A117346 _Walter Nissen_, Mar 09 2006
%E A117346 First term prepended by _Harvey P. Dale_, Dec 25 2013