A119347 Number of distinct sums of distinct divisors of n.

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

1,2

Comments

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics 76 (1954), pp. 779-785.

Formula

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

Example

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.

Maple

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);

Mathematica

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

Prog

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

Crossrefs

Cf. A000203, A002093, A005153, A027750, A030057, A093890, A119348, A225561.

Sequence in context: A143275 A083262 A122978 * A323774 A062402 A347405

Adjacent sequences: A119344 A119345 A119346 * A119348 A119349 A119350

Keyword

nonn

Author

Emeric Deutsch, May 15 2006

Status

approved