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%I A225574 #23 Sep 27 2019 08:44:37
%S A225574 1,3,7,12,15,28,31,39,42,56,60,63,72,90,91,96,120,124,127,144,168,180,
%T A225574 186,195,210,217,224,234,248,252,255,280,312,336,360,363,372,378,392,
%U A225574 399,403,434,465,468,480,504,508,511,546,558,560,576,588,600,620,672,684,702,720,728,744
%N A225574 Additive endpoints: range of A225561.
%C A225574 Numbers n such that 1, 2, ..., n can be represented as the sum of distinct divisors of some number m, but n+1 cannot be so represented.
%C A225574 Note that in the article, the sequence differs at index 17 with term 100 instead of 120. - _Michel Marcus_, Jun 14 2014
%C A225574 Also the range of the sum of divisors function (A000203) over the practical numbers (A005153). The numbers m such that the set of numbers k with A225561(k) = m has a nonvanishing asymptotic density. - _Amiram Eldar_, Sep 27 2019
%H A225574 Amiram Eldar, <a href="/A225574/b225574.txt">Table of n, a(n) for n = 1..10000</a>
%H A225574 Paul Pollack and Lola Thompson, <a href="http://publi.math.unideb.hu/load_jpg.php?p=1792">Practical pretenders</a>, Publicationes Mathematicae Debrecen, Vol. 82, No. 3-4 (2013), pp. 651-717, <a href="http://arxiv.org/abs/1201.3168">arXiv preprint</a>, arXiv:1201.3168 [math.NT], 2012.
%F A225574 Pollack & Thompson show that for each e > 0, n (log n)^(1/e) << a(n) << n^(1+e).
%t A225574 b[n_] := b[n] = First[Complement[Range[DivisorSigma[1, n] + 1], Total /@ Subsets[Divisors[n]]]] - 1; Sort[Tally[Array[b, 300]]][[All, 1]] (* _Jean-François Alcover_, Sep 27 2018 *)
%t A225574 m = 1000; f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most @ fct]), _?(# > 1 &)]) == {}; prac = Select[Range[m], pracQ]; Union @ Select[DivisorSigma[1, prac], # <= m &] (* _Amiram Eldar_, Sep 27 2019 *)
%Y A225574 Cf. A000203, A005153, A225561.
%K A225574 nonn
%O A225574 1,2
%A A225574 _Charles R Greathouse IV_, May 10 2013
%E A225574 More terms from _Jean-François Alcover_, Sep 27 2018
%E A225574 Missing terms inserted by _Amiram Eldar_, Sep 27 2019

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