A100587 -id:A100587 - cp's OEIS frontend

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

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

A229335 Sum of sums of elements of subsets of divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 96, 16, 120, 52, 144, 24, 896, 28, 192, 192, 496, 36, 1248, 40, 1344, 256, 288, 48, 7680, 124, 336, 320, 1792, 60, 9216, 64, 2016, 384, 432, 384, 23296, 76, 480, 448, 11520, 84, 12288, 88, 2688, 2496, 576, 96, 63488, 228, 2976, 576, 3136, 108

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Number of nonempty subsets of divisors of n = A100587(n).

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of sums of elements of subsets = 1 + 2 + 4 + 3 + 5 + 6 + 7 = 28 = (2^3-1) * 2^2 = 7 * 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A229336 (product of sums of elements of subsets of divisors of n).
Cf. A229337 (sum of products of elements of subsets of divisors of n).
Cf. A229338 (product of products of elements of subsets of divisors of n).

A229335 := proc(n)
    numtheory[sigma](n)*A100577(n) ;
end proc:
seq(A229335(n),n=1..100) ; # R. J. Mathar, Nov 10 2017

Table[Total[Flatten[Subsets[Divisors[n]]]], {n, 100}] (* T. D. Noe, Sep 21 2013 *)

a(n) = A000203(n) * A100577(n) = A000203(n) * (A100587(n) + 1) / 2 = A000203(n) * 2^(A000005(n) - 1) = sigma(n) * 2^(tau(n) - 1).

a(2^n) = (2^(n+1) - 1) * 2^n.

A339665 Number of nonempty subsets of divisors of n whose harmonic mean is an integer.

Original entry on oeis.org

1, 2, 2, 3, 2, 9, 2, 4, 3, 4, 2, 17, 2, 4, 6, 5, 2, 19, 2, 10, 4, 4, 2, 37, 3, 4, 4, 12, 2, 45, 2, 6, 4, 4, 4, 57, 2, 4, 4, 28, 2, 29, 2, 6, 16, 4, 2, 85, 3, 6, 4, 6, 2, 35, 4, 23, 4, 4, 2, 301, 2, 4, 6, 7, 4, 28, 2, 6, 4, 19, 2, 255, 2, 4, 10, 6, 4, 20, 2, 61

Offset: 1

Ilya Gutkovskiy, Dec 11 2020

nonn

			a(6) = 9 subsets: {1}, {2}, {3}, {6}, {2, 6}, {3, 6}, {1, 3, 6}, {2, 3, 6} and {1, 2, 3, 6}.

Chai Wah Wu, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Harmonic Mean
Index entries for sequences related to divisors of numbers

Cf. A027750, A100587, A339453, A339663, A339664, A339666.

a[n_] := Count[Subsets[Divisors[n]], ?(Length[#] > 0 && IntegerQ[HarmonicMean[#]] &)]; Array[a, 100] (* _Amiram Eldar, Nov 09 2021 *)

h(s, d) = #s/sum(k=1, #s, 1/d[s[k]]);
a(n) = my(d=divisors(n), nb=0); forsubset(#d, s, if (#s && (denominator(h(s, d))==1), nb++)); nb; \\ Michel Marcus, Dec 15 2020

from itertools import combinations
from sympy import divisors
def A339665(n):
    ds = tuple(divisors(n, generator=True))
    return sum(sum(1 for d in combinations(ds,i) if n*i % sum(d) == 0) for i in range(1,len(ds)+1)) # Chai Wah Wu, Nov 09 2021

A100577 Number of sets of divisors of n with an odd sum.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 8, 4, 8, 2, 32, 2, 8, 8, 16, 2, 32, 2, 32, 8, 8, 2, 128, 4, 8, 8, 32, 2, 128, 2, 32, 8, 8, 8, 256, 2, 8, 8, 128, 2, 128, 2, 32, 32, 8, 2, 512, 4, 32, 8, 32, 2, 128, 8, 128, 8, 8, 2, 2048, 2, 8, 32, 64, 8, 128, 2, 32, 8, 128, 2, 2048, 2, 8, 32, 32, 8, 128, 2, 512, 16, 8, 2

Offset: 1

Reinhard Zumkeller, Nov 29 2004

a(n) = A000079(A032741(n)).

Also number of subsets of divisors of n which do not contain 1; thus a(n) = (A100587(n)+1)/2. - Vladeta Jovovic, Jul 02 2007

			a(12) = #{{1}, {3}, {1,2}, {1,4}, {2,3}, {1,6}, {3,4}, {1,2,4}, {3,6}, {1,2,6}, {2,3,4}, {1,4,6}, {2,3,6}, {1,12}, {3,4,6}, {1,2,4,6}, {3,12}, {1,2,12}, {2,3,4,6}, {1,4,12}, {2,3,12}, {1,6,12}, {3,4,12}, {1,2,4,12}, {3,6,12}, {1,2,6,12}, {2,3,4,12}, {1,4,6,12}, {2,3,6,12}, {1,2,4,6,12}, {3,4,6,12}, {2,3,4,6,12}} = 32.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000079, A001227, A032741, A100587.

A100577 := proc(n)
    2^(numtheory[tau](n)-1) ;
end proc:
seq(A100577(n),n=1..100) ; # R. J. Mathar, Nov 10 2017

Table[2^(DivisorSigma[0, n] - 1), {n, 1, 100}] (* Jean-François Alcover, Feb 13 2018 *)

a(n)=2^(numdiv(n)-1) \\ Charles R Greathouse IV, Jan 19 2017

a(n) = 2^(A000005(n)-1).

A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 9, 0, 20, 0, 39, 27, 54, 0, 77, 108, 108, 0, 135, 0, 170, 0, 272, 378, 252, 0, 372, 567, 500, 0, 405, 0, 464, 0, 792, 1053, 792, 0, 665, 1350, 1148, 0, 819, 0, 902, 882, 897, 2052, 1080, 0, 1425, 1395, 2052, 1715, 1377, 0, 2052, 0, 2600, 3375, 1710

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 2 + 3 + 4 = 9 (numbers 2, 3 and 4 are not a sum of any subset of distinct divisors of 5).
Numbers n = 14 and 15 are an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237290(14) = A237290(15).
a(14) = 4+5+6+11+12+13+18+19+20 = a(15) = 2+7+10+11+12+13+14+17+22 = 108.
a(6) = 0 as 6 is practical; the sums into distinct divisors from 1 through 12 are 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 2 + 3, 7 through 12 are (1 through 6) + 6. So none are not a sum distinct divisors of 6. - _David A. Corneth_, Jul 22 2025

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A119348, A005153, A184387, A237287, A237290.

isSumDist := proc(n,k)
    local dvs ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237289 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if not isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
    a ;
end proc:
seq(A237289(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Block[{d = Divisors@n, s}, s = Plus @@ d; s*(s + 1)/2 - Plus @@ Union[Plus @@@ Subsets@d]]; m = Array[a, 59] (* Giovanni Resta, Mar 13 2014 *)

from sympy import divisors
def A237289(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return (s*(s+1)>>1)-sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237290(n).
a(p) = p(p - 1) / 2 - 1 for p = prime > 2.
a(n) = 0 for practical numbers (A005153), a(n) > 0 for numbers that are not practical (A237287).
a(n) = A184387(n) - A229335(n) for numbers n such that A119347(n) = A100587(n).

a(55) and a(57)-a(59) corrected by Giovanni Resta, Mar 13 2014

A339663 Number of nonempty subsets of divisors of n whose average is an integer.

Original entry on oeis.org

1, 2, 3, 4, 3, 9, 3, 7, 6, 6, 3, 24, 3, 7, 13, 12, 3, 27, 3, 22, 11, 7, 3, 72, 6, 6, 12, 21, 3, 83, 3, 20, 13, 6, 11, 133, 3, 7, 11, 70, 3, 82, 3, 21, 38, 7, 3, 230, 7, 14, 13, 23, 3, 88, 11, 65, 11, 6, 3, 763, 3, 7, 35, 36, 11, 84, 3, 22, 13, 73, 3, 780, 3, 6, 37, 20, 11, 82, 3, 228

Offset: 1

Ilya Gutkovskiy, Dec 11 2020

nonn

			a(16) = 12 subsets: {1}, {2}, {4}, {8}, {16}, {2, 4}, {2, 8}, {2, 16}, {4, 8}, {4, 16}, {8, 16} and {1, 4, 16}.

Antti Karttunen, Table of n, a(n) for n = 1..1079
Eric Weisstein's World of Mathematics, Arithmetic Mean
Index entries for sequences related to divisors of numbers

Cf. A027750, A051293, A100587, A339664, A339665, A339666.

sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };
A339663(n) = { my(ds=divisors(n), u=#ds); sum(m=1, (2^u)-1, !(sumbybits(ds,m)%hammingweight(m))); }; \\ Antti Karttunen, Dec 12 2021

A339666 Number of nonempty subsets of divisors of n whose root-mean-square is an integer.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 6, 2, 6, 4, 5, 2, 6, 2, 6, 6, 4, 2, 8, 3, 4, 4, 9, 2, 8, 2, 6, 4, 4, 7, 9, 2, 4, 4, 12, 3, 12, 2, 6, 7, 4, 2, 12, 5, 6, 4, 6, 2, 8, 5, 12, 4, 4, 2, 26, 2, 4, 9, 7, 4, 8, 2, 6, 4, 14, 2, 12, 2, 4, 6, 6, 6, 8, 2, 24, 5, 6, 2, 22

Offset: 1

Ilya Gutkovskiy, Dec 11 2020

nonn

			a(14) = 6 subsets: {1}, {2}, {7}, {14}, {1, 7} and {2, 14}.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Root-Mean-Square
Index entries for sequences related to divisors of numbers

Cf. A027750, A100587, A339454, A339663, A339664, A339665.

a:= proc(n) option remember; uses numtheory; local l, b;
      l, b:= sort([divisors(n)[]]),
      proc(i, s, c) option remember;
        `if`(i=0, `if`(c>0 and issqr(s/c), 1, 0),
         b(i-1, s, c)+b(i-1, s+l[i]^2, c+1))
      end; forget(b); b(nops(l), 0$2)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 30 2022

a[n_] := a[n] = Module[{b, l = Divisors[n]}, b[i_, s_, c_] := b[i, s, c] = If[i == 0, If[c > 0 && IntegerQ @ Sqrt[s/c], 1, 0], b[i-1, s, c]+b[i-1, s+l[[i]]^2, c+1]]; b[Length[l], 0, 0]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 13 2022, after Alois P. Heinz *)

A229336 Product of sums of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 6, 12, 5040, 30, 77598259200, 56, 1307674368000, 168480, 12703122432000, 132, 52875224823823084892891318660312910903645116196873830400000000000000, 182, 440505199411200, 493242753024000, 8222838654177922817725562880000000, 306

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Number of nonempty subsets of divisors of n = A100587(n).

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of sums of elements of subsets = 1*2*4*3*5*6*7 = 5040 = (2^3 - 1)! = 7!.

Cf. A229335 (sum of sums of elements of nonempty subsets of divisors of n),
A229337 (sum of products of elements of nonempty subsets of divisors of n),
A229338 (product of products of elements of nonempty subsets of divisors of n).

Table[Times@@(Total/@Rest[Subsets[Divisors[n]]]),{n,20}] (* Harvey P. Dale, Jan 22 2023 *)

a(2^n) = (2^(n+1) - 1)!.

A229337 Sum of products of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 5, 7, 29, 11, 167, 15, 269, 79, 395, 23, 10919, 27, 719, 767, 4589, 35, 31919, 39, 41579, 1407, 1655, 47, 2456999, 311, 2267, 2239, 104399, 59, 5499647, 63, 151469, 3263, 3779, 3455, 76767599, 75, 4679, 4479, 15343019, 83, 19071359, 87, 372599, 353279, 6767

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Number of nonempty subsets of divisors of n = A100587(n).

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of products of elements of subsets = 1 + 2 + 4 + 2 + 4 + 8 + 8 = 29 = (1+1) * (2+1) * (4+1) - 1.

Cf. A229335 (sum of sums of elements of nonempty subsets of divisors of n), A229336 (product of sums of elements of nonempty subsets of divisors of n), A229338 (product of products of elements of nonempty subsets of divisors of n).

Let a, b, c, ..., k be all divisors of n; a(n) = (a+1) * (b+1) * ... * (k+1) - 1.
a(p) = 2p+1, a(p^2) = 2(p+1)(p^2+1) - 1.
a(n) = A020696(n) - 1.

A339664 Number of nonempty subsets of divisors of n whose geometric mean is an integer.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 8, 5, 4, 2, 10, 2, 4, 4, 15, 2, 10, 2, 10, 4, 4, 2, 16, 5, 4, 8, 10, 2, 8, 2, 26, 4, 4, 4, 63, 2, 4, 4, 16, 2, 8, 2, 10, 10, 4, 2, 30, 5, 10, 4, 10, 2, 16, 4, 16, 4, 4, 2, 20, 2, 4, 10, 45, 4, 8, 2, 10, 4, 8, 2, 196, 2, 4, 10, 10, 4, 8, 2, 30

Offset: 1

Ilya Gutkovskiy, Dec 11 2020

nonn

			a(12) = 10 subsets: {1}, {2}, {3}, {4}, {6}, {12}, {1, 4}, {3, 12}, {1, 2, 4} and {3, 6, 12}.

Eric Weisstein's World of Mathematics, Geometric Mean
Index entries for sequences related to divisors of numbers

Cf. A027750, A100587, A326027, A339663, A339665, A339666.

cp's OEIS Frontend

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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A229335 Sum of sums of elements of subsets of divisors of n.

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A339665 Number of nonempty subsets of divisors of n whose harmonic mean is an integer.

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A100577 Number of sets of divisors of n with an odd sum.

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A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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A339663 Number of nonempty subsets of divisors of n whose average is an integer.

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A339666 Number of nonempty subsets of divisors of n whose root-mean-square is an integer.

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