A378450 -id:A378450 - cp's OEIS frontend

A378597 Möbius transform of A378450, where A378450 is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

0, 0, 1, 0, 3, -1, 5, 0, 5, 0, 9, 0, 11, 4, 5, 0, 15, -5, 17, -3, 11, 12, 21, 0, 21, 16, 19, -9, 27, -8, 29, 0, 23, 24, 25, 0, 35, 28, 29, 0, 39, -20, 41, 0, 9, 36, 45, 0, 45, 6, 41, 8, 51, -19, 45, 0, 47, 48, 57, 3, 59, 52, 23, 0, 55, -44, 65, 24, 59, -35, 69, 0, 71, 64, 31, 32, 67, -56, 77, 0, 65, 72, 81, 9, 75, 76, 77, -21

Offset: 1

Antti Karttunen, Dec 02 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008683, A119347, A378450, A378596.

A119347(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; };
A378450(n) = (sigma(n)-A119347(n));
A378597(n) = sumdiv(n,d,moebius(n/d)*A378450(d));

a(n) = Sum_{d|n} A008683(d)*A378450(n/d).

a(n) = n - A378596(n).

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

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A378597 Möbius transform of A378450, where A378450 is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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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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