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A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

%I A377929 #27 Dec 30 2024 02:24:10
%S A377929 1,2,3,4,5,6,7,8,11,12,13,16,17,18,19,20,23,24,28,29,30,31,32,36,37,
%T A377929 40,41,42,43,47,48,53,54,56,59,60,61,64,66,67,71,72,73,78,79,80,83,84,
%U A377929 88,89,90,96,97,100,101,103,104,107,108,109,112,113,120,126,127,128
%N A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.
%C A377929 Equivalently, positive integers m such that every number k <= d is a sum of distinct proper divisors of m, where d is the largest proper divisor of m (follows from Corollary 2.11 in the Kukla and Miska paper).
%C A377929 Rao and Peng (2013) proved that a number is quasi practical if and only if it is prime or practical (also Theorem 2.9 in Kukla/Miska paper).
%H A377929 Andrzej Kukla, <a href="/A377929/b377929.txt">Table of n, a(n) for n = 1..10000</a>
%H A377929 K. P. S. Bhaskara Rao and Yuejian Peng, <a href="https://doi.org/10.1016/j.jnt.2012.09.020">On Zumkeller numbers</a>, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
%H A377929 Andrzej Kukla and Piotr Miska, <a href="https://arxiv.org/abs/2405.18225">On practical sets and A-practical numbers</a>, arXiv:2405.18225 [math.NT], 2024.
%t A377929 QuasiPracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]] || PrimeQ[n]]; Select[Range[200], QuasiPracticalQ] (* Created based on code by T. D. Noe, Apr 02 2010 *)
%Y A377929 Union of A000040 and A005153.
%Y A377929 Cf. A002093, A103288, A033630, A119348, A174533, A174973, A083207.
%K A377929 nonn,easy
%O A377929 1,2
%A A377929 _Andrzej Kukla_, Nov 11 2024