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

1,2

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.

Formula

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

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 *)

Crossrefs

Cf. A000203, A005153, A225561.

Sequence in context: A296094 A075895 A033015 * A317305 A356425 A096998

Adjacent sequences: A225571 A225572 A225573 * A225575 A225576 A225577

Keyword

nonn

Author

Charles R Greathouse IV, May 10 2013

Extensions

More terms from Jean-François Alcover, Sep 27 2018

Missing terms inserted by Amiram Eldar, Sep 27 2019

Status

approved