A225574 Additive endpoints: range of A225561.

1, 3, 7, 12, 15, 28, 31, 39, 42, 56, 60, 63, 72, 90, 91, 96, 120, 124, 127, 144, 168, 180, 186, 195, 210, 217, 224, 234, 248, 252, 255, 280, 312, 336, 360, 363, 372, 378, 392, 399, 403, 434, 465, 468, 480, 504, 508, 511, 546, 558, 560, 576, 588, 600, 620, 672, 684, 702, 720, 728, 744

Offset: 1

Author

Charles R Greathouse IV, May 10 2013

Keywords

• nonn

Comments

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.

Note that in the article, the sequence differs at index 17 with term 100 instead of 120. - Michel Marcus, Jun 14 2014

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Pollack and Lola Thompson, Practical pretenders, Publicationes Mathematicae Debrecen, Vol. 82, No. 3-4 (2013), pp. 651-717, arXiv preprint, arXiv:1201.3168 [math.NT], 2012.

Crossrefs

Cf. A000203, A005153, A225561.

Programs

Mathematica

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 *)
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 *)

Formula

Pollack & Thompson show that for each e > 0, n (log n)^(1/e) << a(n) << n^(1+e).

Extensions

More terms from Jean-François Alcover, Sep 27 2018

Missing terms inserted by Amiram Eldar, Sep 27 2019