

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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


REFERENCES

B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics 76 (1954), pp. 779785.


LINKS

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


FORMULA

For n > 1, 3 <= a(n) <= sigma(n).  Charles R Greathouse IV, Feb 11 2019


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^t1)}) end: seq(a(n), n=1..90);


MATHEMATICA

a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
Array[a, 100] (* JeanFrançois 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


CROSSREFS

Cf. A000203, A002093, A005153, A027750, A030057, A093890, A119348, A225561.
Sequence in context: A143275 A083262 A122978 * A323774 A062402 A294015
Adjacent sequences: A119344 A119345 A119346 * A119348 A119349 A119350


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 15 2006


STATUS

approved



