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A119347
Number of distinct sums of distinct divisors of n.
13
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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 15 2006
STATUS
approved