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A225574 Additive endpoints: range of A225561. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A096998 A317307

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

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)