A225561 -id:A225561 - cp's OEIS frontend

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

Charles R Greathouse IV, May 10 2013

nonn

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

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.

Cf. A000203, A005153, A225561.

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 *)

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

More terms from Jean-François Alcover, Sep 27 2018

Missing terms inserted by Amiram Eldar, Sep 27 2019

A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

Original entry on oeis.org

1, 3, 3, 7, 3, 12, 3, 15, 7, 15, 3, 28, 3, 15, 15, 31, 3, 39, 3, 42, 15, 15, 3, 60, 7, 15, 15, 56, 3, 72, 3, 63, 15, 15, 15, 91, 3, 15, 15, 90, 3, 96, 3, 63, 55, 15, 3, 124, 7, 63, 15, 63, 3, 120, 15, 120, 15, 15, 3, 168, 3, 15, 59, 127, 15, 144, 3, 63, 15, 142, 3, 195, 3, 15, 63, 63

Offset: 1

Emeric Deutsch, May 15 2006

nonn

If a(n)=sigma(n) (=sum of the divisors of n =A000203(n); i.e. all numbers from 1 to sigma(n) are sums of distinct divisors of n), then n is called a practical number (A005153). The actual sums obtained from the divisors of n are given in row n of the triangle A119348.
The records appear to occur at the highly abundant numbers, A002093, excluding 3 and 10. For n in A174533, a(n) = sigma(n)-2. - T. D. Noe, Mar 29 2010
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020
Zumkeller numbers A083207 give the positions of even terms in this sequence (likewise, the positions of odd terms in A308605). - Antti Karttunen and Ilya Gutkovskiy, Nov 29 2024

			a(5)=3 because the divisors of 5 are 1 and 5 and all the possible sums: are 1,5 and 6; a(6)=12 because we can form all sums 1,2,...,12 by adding up the terms of a nonempty subset of the divisors 1,2,3,6 of 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Jon Maiga, Computer-generated formulas for A119347, Sequence Machine.
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics 76 (1954), pp. 779-785.

One less than A308605.
Cf. A000203, A002093, A005153, A027750, A030057, A033630, A093890, A100587, A119348, A193279, A225561, A237290, A378447, A378450, A378451.
Cf. A083207 (positions of even terms).

import Data.List (subsequences, nub)
a119347 = length . nub . map sum . tail . subsequences . a027750_row'
-- Reinhard Zumkeller, Jun 27 2015

with(numtheory): with(linalg): a:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: seq(a(n),n=1..90);

a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Jan 27 2018 *)

A119347(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=1,poldegree(p),(0Antti Karttunen, Nov 28 2024

A119347(n) = { my(c=[0]); fordiv(n, d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A119347(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    return len(c)-1 # Chai Wah Wu, Jul 05 2023

For n > 1, 3 <= a(n) <= sigma(n). - Charles R Greathouse IV, Feb 11 2019
For p prime, a(p) = 3. For k >= 0, a(2^k) = 2^(k + 1) - 1. - Ctibor O. Zizka, Oct 19 2023
From Antti Karttunen, Nov 29 2024: (Start)
a(n) = A308605(n)-1.
a(n) = 2*(A237290(n)/A000203(n)) - 1. [Found by Sequence Machine. See A237290.]
a(n) <= A100587(n).
(End)

Definition clarified by Antti Karttunen, Nov 29 2024

A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 29 2019

T(n, k) > 0 for all values of k iff n is practical (A005153).

			Table begins as:
  1
  1,1,1
  1,0,1,1
  1,1,1,1,1,1,1
  1,0,0,0,1,1
  1,1,2,1,1,2,1,1,2,1,1,1
  1,0,0,0,0,0,1,1
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  1,0,1,1,0,0,0,0,1,1,0,1,1
  1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1

Amiram Eldar, Table of n, a(n) for n = 1..8299 (rows 1..100 flattened)

Cf. A000203 (row lengths), A307224 (row products).

Cf. A005153, A027750, A030057, A033630, A119348, A225561, A237287, A322860.

T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; Table[T[n, k], {n,1,10}, {k, 1, DivisorSigma[1,n]}] // Flatten

T(n, n) = A033630(n).

T(n, A030057(n)) = 0 if there is a 0 in the n-th row, i.e. A030057(n) <= sigma(n) or n is not practical.

A327832 The practical component of n: the largest divisor of n which is a practical number (A005153).

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 2, 1, 24, 1, 2, 1, 28, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 56, 1, 2, 1, 60, 1, 2, 1, 64, 1, 66, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 78, 1, 80, 1

Offset: 1

Amiram Eldar, Sep 27 2019

nonn

From Andreas Weingartner, Jun 30 2021: (Start)
Let r_m be the natural density of the set of integers n with a(n) = m. Then r_m is positive if and only if m is practical. In that case, r_m = (1/m)*P_m, where P_m is the product of (1-1/p) over primes p <= sigma(m) + 1 (see Cor. 1 of Weingartner 2015). The first few values of (m, r_m) are (1, 1/2), (2, 1/6), (4, 2/35), (6, 32/1001), (8, 24/1001), (12, 36864/2800733), ...
As y grows, the natural density of integers n, which satisfy a(n) > y, is asymptotic to c*exp(-gamma)/log(y), where c = 1.33607... is the constant factor in the asymptotic for the count of practical numbers (A005153) and gamma = 0.577215... is Euler's constant (see Eq. (3) of Weingartner (2015)). For example, about 1% of integers n satisfy a(n) > exp(75), because c*exp(-gamma)/75 = 0.010... (End)

			a(22) = 2 since the divisors of 22 are {1, 2, 11, 22}, of them {1, 2} are practical, and 2 being the largest.

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.
Andreas Weingartner, Integers with large practical component, Publicationes Mathematicae Debrecen, Vol. 87, No. 3-4 (2015), pp. 439-447, arXiv preprint, arXiv:1411.6974v2 [math.NT], 2014-2015.

Cf. A005153, A225561, A225574.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[n_] := If[(ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}, n, Times @@ (Power @@@ fct[[1 ;; ind[[1, 1]] - 1]])]; Array[a, 100]

\\ using is_A005153
a(n) = fordiv(n, d, if(is_A005153(n/d), return(n/d))); \\ Michel Marcus, Jul 03 2021

If n = Product_{i=1..r} p_i^e_i, then define n_0 = 1, n_j = Product_{i=1..j} p_i^e_i. a(n) = n_j where j is the first index for which p_{j+1} > sigma(n_j) + 1, or j = r if no such index exists.
A number n is practical if and only if a(n) = n.
a(n) = 1 if and only if n is odd.
A000203(a(n)) = A225561(n).

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A225574 Additive endpoints: range of A225561.

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A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

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A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

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A327832 The practical component of n: the largest divisor of n which is a practical number (A005153).

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