A294137 -id:A294137 - cp's OEIS frontend

A294138 Number of compositions (ordered partitions) of n into proper divisors of n.

1, 0, 1, 1, 5, 1, 24, 1, 55, 19, 128, 1, 1627, 1, 741, 449, 5271, 1, 45315, 1, 83343, 3320, 29966, 1, 5105721, 571, 200389, 26425, 5469758, 1, 154004510, 1, 47350055, 226019, 9262156, 51885, 15140335649, 1, 63346597, 2044894, 14700095925, 1, 185493291000, 1, 35539518745, 478164162

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(4) = 5 because 4 has 3 divisors {1, 2, 4} among which 2 are proper divisors {1, 2} therefore we have [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Amiram Eldar, Table of n, a(n) for n = 0..3359
Index entries for sequences related to compositions

Cf. A027750, A032741, A065205, A100346, A210442, A294137.

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[d[[k]] != n] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 45}]

a(n) = [x^n] 1/(1 - Sum_{d|n, d < n} x^d).

a(n) = A100346(n) - 1.

A331979 Number of compositions (ordered partitions) of n into distinct nontrivial divisors of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 894, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 894, 0, 0, 0, 0, 0, 30, 0, 120, 0, 0, 0, 19518, 0, 0, 0, 0, 0, 126, 0, 0, 0, 0, 0, 18558, 0, 0, 0, 0, 0, 6, 0, 864

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(12) = 6 because we have [6, 4, 2], [6, 2, 4], [4, 6, 2], [4, 2, 6], [2, 6, 4] and [2, 4, 6].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A018818, A027750, A033630, A065205, A070824, A100346, A211111, A293813, A293814, A294137, A294138, A331927, A331928.

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
      b:= proc(m, i, p) option remember; `if`(m=0, p!, `if`(i<1, 0,
             b(m, i-1, p)+`if`(l[i]>m, 0, b(m-l[i], i-1, p+1))))
          end; forget(b):
      b(n, nops(l), 0)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 03 2020

a[n_] := If[n == 0, 1, Module[{b, l = Divisors[n] ~Complement~ {1, n}}, b[m_, i_, p_] := b[m, i, p] = If[m == 0, p!, If[i < 1, 0, b[m, i-1, p] + If[l[[i]] > m, 0, b[m - l[[i]], i-1, p+1]]]]; b[n, Length[l], 0]]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A357312 Number of compositions (ordered partitions) of n into divisors of n that are smaller than sqrt(n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 13, 1, 34, 1, 89, 1, 927, 1, 610, 189, 1597, 1, 35890, 1, 46754, 1873, 28657, 1, 3919944, 1, 196418, 18560, 4205249, 1, 110187694, 1, 39882198, 183916, 9227465, 9496, 10312882481, 1, 63245986, 1822473, 11969319436, 1, 141930520462, 1, 34020543362, 339200673

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A100346, A294137, A294138, A327766, A357311.

a:= proc(n) option remember; uses numtheory; local b, l;
      l, b:= select(x-> is(xm, 0, b(m-j)), j=l))
      end; b(n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 23 2022

a[n_] := SeriesCoefficient[1/(1 - Sum[Boole[d < Sqrt[n]] x^d, {d, Divisors[n]}]), {x, 0, n}]; Table[a[n], {n, 0, 45}]

a(n) = [x^n] 1 / (1 - Sum_{d|n, d < sqrt(n)} x^d).

A327766 Number of compositions (ordered partitions) of n into divisors of n that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 13, 1, 34, 19, 89, 1, 927, 1, 610, 189, 4930, 1, 35890, 1, 46754, 1873, 28657, 1, 3919944, 571, 196418, 18560, 4205249, 1, 110187694, 1, 39882198, 183916, 9227465, 9496, 14484956252, 1, 63245986, 1822473, 11969319436, 1, 141930520462, 1, 34020543362, 339200673

Offset: 0

Ilya Gutkovskiy, Sep 24 2019

nonn

Cf. A100346, A294137, A294138, A327642.

a[n_] := SeriesCoefficient[1/(1 - Sum[Boole[d <= Sqrt[n]] x^d, {d, Divisors[n]}]), {x, 0, n}]; Table[a[n], {n, 0, 45}]

a(n) = [x^n] 1 / (1 - Sum_{d|n, d <= sqrt(n)} x^d).

a(p) = 1, where p is prime.

cp's OEIS Frontend

A294138 Number of compositions (ordered partitions) of n into proper divisors of n.

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A331979 Number of compositions (ordered partitions) of n into distinct nontrivial divisors of n.

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A357312 Number of compositions (ordered partitions) of n into divisors of n that are smaller than sqrt(n).

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Maple

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A327766 Number of compositions (ordered partitions) of n into divisors of n that are at most sqrt(n).

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