A171565 -id:A171565 - cp's OEIS frontend

A018818 Number of partitions of n into divisors of n.

1, 2, 2, 4, 2, 8, 2, 10, 5, 11, 2, 45, 2, 14, 14, 36, 2, 81, 2, 92, 18, 20, 2, 458, 7, 23, 23, 156, 2, 742, 2, 202, 26, 29, 26, 2234, 2, 32, 30, 1370, 2, 1654, 2, 337, 286, 38, 2, 9676, 9, 407, 38, 454, 2, 3132, 38, 3065, 42, 47, 2, 73155, 2, 50, 493, 1828, 44, 5257, 2, 740, 50, 5066

Offset: 1

David W. Wilson

From Reinhard Zumkeller, Dec 11 2009: (Start)
For odd primes p: a(p^2) = p + 2; for n > 1: a(A001248(n)) = A052147(n);
For odd primes p > 3, a(3*p) = 2*p + 4; for n > 2: a(A001748(n)) = A100484(n) + 4. (End)
From Matthew Crawford, Jan 19 2021: (Start)
For a prime p, a(p^3) = (p^3 + p^2 + 2*p + 4)/2;
For distinct primes p and q, a(p*q) = (p+1)*(q+1)/2 + 2. (End)

			The a(6) = 8 representations of 6 are 6 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Douglas Bowman et al., Problem 6640: Partitions of n into parts which are divisors of n, American Mathematical Monthly, 99(3) (1992), 276-277.
Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math., 6(11) (1975), 1276-1286.
Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math., 6(11) (1975), 1276-1286.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 1.
Rémy Sigrist, Colored logarithmic scatterplot of the first 10000 terms (where the color is function of A000005(n)).
Index entries for sequences related to partitions.

Cf. A002577, A027750, A033630, A072721, A161148 (partitions in squared divisors), A171565, A210442, A211110, A225244, A306387, A379957.

Cf. A000005, A001248, A001748, A052147, A100484.

a018818 n = p (init $ a027750_row n) n + 1 where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 02 2012

[#RestrictedPartitions(n,{d:d in Divisors(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019

A018818 := proc(n)
    local a,p,w,el ;
    a := 0 ;
    for p in combinat[partition](n) do
        w := true ;
        for el in p do
            if modp(n,el) <> 0 then
                w := false;
                break;
            end if;
        end do:
        if w then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[1/Product[1 - x^d[[i]], {i, Length[d]}], {x, 0, n}], x, n], {n, 100}] (* T. D. Noe, Jul 28 2011 *)

a(n)=numbpartUsing(n, divisors(n));
numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1,mx, numbpartUsing(n-v[i],v,i)) \\ inefficient; Charles R Greathouse IV, Jun 21 2017

A018818(n) = { my(p = Ser(1, 'x, 1+n)); fordiv(n, d, p /= (1 - 'x^d)); polcoef(p, n); }; \\ Antti Karttunen, Jan 23 2025, after Vladeta Jovovic

Coefficient of x^n in the expansion of 1/Product_{d|n} (1-x^d). - Vladeta Jovovic, Sep 28 2002
a(n) = 2 iff n is prime. - Juhani Heino, Aug 27 2009
a(n) = f(n,n,1), where f(n,m,k) = f(n,m,k+1) + f(n,m-k,k)*0^(n mod k) if k <= m, otherwise 0^m. - Reinhard Zumkeller, Dec 11 2009
Paul Erdős, Andrew M. Odlyzko, and the Editors of the AMM give bounds; see Bowman et al. - Charles R Greathouse IV, Dec 04 2012

A182469 Triangle read by rows in which row n lists the odd divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 3, 5, 15, 1, 1, 17, 1, 3, 9, 1, 19, 1, 5, 1, 3, 7, 21, 1, 11, 1, 23, 1, 3, 1, 5, 25, 1, 13, 1, 3, 9, 27, 1, 7, 1, 29, 1, 3, 5, 15, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 5, 7, 35, 1

Offset: 1

Reinhard Zumkeller, Apr 30 2012

n-th row = intersection of A005408 and of n-th row of A027750.

			The triangle begins:
.  1   {1}
.  2   {1}
.  3   {1,3}
.  4   {1}
.  5   {1,5}
.  6   {1,3}
.  7   {1,7}
.  8   {1}
.  9   {1,3,9}
. 10   {1,5}
. 11   {1,11}
. 12   {1,3}
. 13   {1,13}
. 14   {1,7}
. 15   {1,3,5,15}
. 16   {1} .

Reinhard Zumkeller, Rows n = 1..2500 of triangle, flattened

Cf. A001227 (row lengths), A000593 (row sums), A136655 (row products).
Cf. A000265, A005408, A027750, A050999, A051000, A037283, A037284, A037285, A171565.
Cf. also A237048.

a182469 n k = a182469_tabf !! (n-1) !! (k-1)
a182469_row = a027750_row . a000265
a182469_tabf = map a182469_row [1..]

Flatten[Table[Select[Divisors[n],OddQ],{n,40}]] (* Harvey P. Dale, Aug 13 2012 *)
Flatten[Table[Divisors[n / 2^IntegerExponent[n, 2]], {n, 40}]] (* Amiram Eldar, May 02 2025 *)

tabf(nn) = {for (n=1, nn, fordiv(n, d, if (d%2, print1(d, ", "))); print(););} \\ Michel Marcus, Apr 22 2017

row(n) = divisors(n >> valuation(n, 2)); \\ Amiram Eldar, May 02 2025

from sympy import divisors
def row(n):
    return [d for d in divisors(n) if d % 2]
for n in range(1, 21): print(row(n)) # Indranil Ghosh, Apr 22 2017

T(n,k) = A027750(A000265(n),k), 1 <= k <= A001227(n).

A000265(n) = T(n,A001227(n)).

A284466 Number of compositions (ordered partitions) of n into odd divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 2, 1, 20, 8, 2, 60, 2, 10, 450, 1, 2, 726, 2, 140, 3321, 14, 2, 5896, 572, 16, 26426, 264, 2, 394406, 2, 1, 226020, 20, 51886, 961584, 2, 22, 2044895, 38740, 2, 20959503, 2, 676, 478164163, 26, 2, 56849086, 31201, 652968, 184947044, 980, 2, 1273706934, 6620376, 153366, 1803937344

Offset: 0

Ilya Gutkovskiy, Mar 27 2017

nonn

			a(10) = 8 because 10 has 4 divisors {1, 2, 5, 10} among which 2 are odd {1, 5} therefore we have [5, 5], [5, 1, 1, 1, 1, 1], [1, 5, 1, 1, 1, 1], [1, 1, 5, 1, 1, 1], [1, 1, 1, 5, 1, 1], [1, 1, 1, 1, 5, 1], [1, 1, 1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to compositions

Cf. A000045, A005408, A032021, A100346, A171565.

with(numtheory):
a:= proc(n) option remember; local b, l;
      l, b:= select(x-> is(x:: odd), divisors(n)),
      proc(m) option remember; `if`(m=0, 1,
         add(`if`(j>m, 0, b(m-j)), j=l))
      end; b(n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[Mod[d[[k]], 2] == 1] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 57}]

from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def a(n):
    l=[x for x in divisors(n) if x%2]
    @cacheit
    def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
    return b(n)
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 01 2017, after Maple code

a(n) = [x^n] 1/(1 - Sum_{d|n, d positive odd} x^d).
a(n) = 1 if n is a power of 2.
a(n) = 2 if n is an odd prime.

A294141 Number of partitions of n into odd parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 3, 0, 5, 5, 2, 7, 5, 1, 11, 14, 6, 9, 20, 5, 18, 33, 5, 43, 50, 12, 58, 17, 16, 91, 96, 26, 58, 146, 23, 183, 148, 11, 241, 285, 88, 187, 152, 98, 367, 537, 87, 177, 376, 179, 857, 983, 77, 1195, 1274, 79, 1596, 468, 290, 2117, 1887, 549, 460, 3064, 440

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(13) = 2 because we have [7, 3, 3] and [5, 5, 3].

Index entries for sequences related to partitions

Cf. A000009, A098743, A128515, A171565, A200745, A294142.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] > 0 && OddQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 72}]

A294142 Number of partitions of n into distinct odd parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 1, 2, 2, 3, 0, 4, 4, 4, 2, 6, 1, 6, 8, 4, 10, 12, 4, 12, 5, 7, 17, 17, 8, 14, 24, 9, 29, 24, 4, 33, 40, 25, 29, 28, 23, 45, 63, 23, 30, 52, 37, 84, 99, 26, 113, 112, 23, 143, 60, 57, 173, 143, 89, 70, 226, 87, 256, 256, 53, 245, 135, 127, 378, 233

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(14) = 2 because we have [11, 3] and [9, 5].

Index entries for sequences related to partitions

Cf. A000700, A098743, A171565, A200745, A209402, A294141.

Table[SeriesCoefficient[Product[1 + Boole[Mod[n, k] > 0 && OddQ[k]] x^k, {k, 1, n}], {x, 0, n}], {n, 0, 80}]

A327640 Number of partitions of n into divisors d of n such that n/d is odd.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 14, 1, 2, 5, 2, 2, 18, 2, 2, 2, 7, 2, 23, 2, 2, 14, 2, 1, 26, 2, 26, 5, 2, 2, 30, 2, 2, 18, 2, 2, 286, 2, 2, 2, 9, 7, 38, 2, 2, 23, 38, 2, 42, 2, 2, 14, 2, 2, 493, 1, 44, 26, 2, 2, 50, 26, 2, 5, 2, 2, 698, 2, 50, 30, 2, 2

Offset: 0

Ilya Gutkovskiy, Sep 20 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 3001 terms from Michael De Vlieger)

Cf. A018818, A038550 (positions of 2's), A171565.

[1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)|IsOdd(n div d)}):n in [1..80]]; // Marius A. Burtea, Sep 20 2019

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> is((n/x):: odd), divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 06 2021

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[OddQ[n/d]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 80}]

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

a(2^k) = 1.

cp's OEIS Frontend

A018818 Number of partitions of n into divisors of n.

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A182469 Triangle read by rows in which row n lists the odd divisors of n.

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A284466 Number of compositions (ordered partitions) of n into odd divisors of n.

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A294141 Number of partitions of n into odd parts that do not divide n.

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A294142 Number of partitions of n into distinct odd parts that do not divide n.

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A327640 Number of partitions of n into divisors d of n such that n/d is odd.

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