A225244 -id:A225244 - 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

A073576 Number of partitions of n into squarefree parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 47, 60, 76, 96, 120, 150, 185, 228, 280, 342, 416, 504, 608, 731, 877, 1048, 1249, 1484, 1759, 2079, 2452, 2885, 3387, 3968, 4640, 5413, 6304, 7328, 8504, 9852, 11395, 13159, 15172, 17468, 20082, 23056, 26434, 30267

Offset: 0

Vladeta Jovovic, Aug 27 2002

nonn

Euler transform of the absolute values of A008683. - Tilman Neumann, Dec 13 2008

Euler transform of A008966. - Vaclav Kotesovec, Mar 31 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A058647.
Cf. A087188.
Cf. A225244.
Cf. A114374.

a073576 = p a005117_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jun 01 2015

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      abs(mobius(d)), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 05 2015

Table[Length[Select[Boole /@ Thread /@ SquareFreeQ /@ IntegerPartitions[n], FreeQ[#, 0] &]], {n, 48}] (* Jayanta Basu, Jul 02 2013 *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*Abs[MoebiusMu[d]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 10 2015, after Alois P. Heinz *)
nmax = 60; CoefficientList[Series[Exp[Sum[Sum[Abs[MoebiusMu[k]] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

from functools import lru_cache
from sympy import mobius, divisors
@lru_cache(maxsize=None)
def A073576(n): return sum(sum(d*abs(mobius(d)) for d in divisors(i, generator=True))*A073576(n-i) for i in range(1,n+1))//n if n else 1 # Chai Wah Wu, Aug 23 2024

G.f.: 1/Product_{k>0} (1-x^A005117(k)).
a(n) = 1/n*Sum_{k=1..n} A048250(k)*a(n-k).
a(n) = A000041(n) - A114374(n) - A117395(n), n>0. - Reinhard Zumkeller, Mar 11 2006
G.f.: 1 + Sum_{i>=1} mu(i)^2*x^i / Product_{j=1..i} (1 - mu(j)^2*x^j). - Ilya Gutkovskiy, Jun 05 2017
a(n) ~ exp(2*sqrt(n)) / (4*Pi^(3/2)*n^(1/4)). - Vaclav Kotesovec, Mar 24 2018

A225245 Number of partitions of n into distinct squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 0, 0, 1, 3, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 1

Offset: 0

Reinhard Zumkeller, May 05 2013

nonn

a(n) <= A033630(n);
a(n) = A033630(n) iff n is squarefree: a(A005117(n)) = A033630(A005117(n));
a(A225353(n)) = 0; a(A225354(n)) > 0.

			a(2*3)     = a(6)  = #{6, 3+2+1} = 2;
a(2*2*3)   = a(12) = #{6+3+2+1} = 1;
a(2*3*5)   = a(30) = #{30, 15+10+5, 15+10+3+2, 15+6+5+3+1} = 4;
a(2*2*3*5) = a(60) = #{30+15+10+5, 30+15+10+3+2, 30+15+6+5+3+1} = 3;
a(2*3*7)   = a(42) = #{42, 21+14+7, 21+14+6+1} = 3;
a(2*2*3*7) = a(84) = #{42+21+14+7, 42+21+14+6+1} = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (5000 terms from Reinhard Zumkeller)
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. 2.

Cf. A005117, A008683, A033630, A206778, A008966, A225244, A087188, A225353, A225354.

a225245 n = p (a206778_row n) n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

a[n_] := If[n == 0, 1, Coefficient[Product[If[MoebiusMu[d] != 0, 1+x^d, 1], {d, Divisors[n]}], x, n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 08 2021, after Ilya Gutkovskiy *)

a(n) = [x^n] Product_{d|n, mu(d) != 0} (1 + x^d), where mu() is the Moebius function (A008683). - Ilya Gutkovskiy, Jul 26 2017

A284345 Number of partitions of n into squares dividing n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 6, 1, 3, 1, 6, 1, 1, 1, 7, 2, 1, 4, 8, 1, 1, 1, 15, 1, 1, 1, 27, 1, 1, 1, 11, 1, 1, 1, 12, 6, 1, 1, 28, 2, 3, 1, 14, 1, 7, 1, 15, 1, 1, 1, 16, 1, 1, 8, 46, 1, 1, 1, 18, 1, 1, 1, 114, 1, 1, 4, 20, 1, 1, 1, 66, 11, 1, 1, 22, 1, 1, 1, 23, 1, 11, 1, 24, 1, 1, 1, 91, 1, 3, 12, 67

Offset: 0

Ilya Gutkovskiy, Mar 25 2017

nonn

			a(8) = 3 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are squares {1, 4} therefore we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A001156, A018818, A046951, A066882, A161148, A225244, A284289.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:=
      sort(select(issqr, [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, Mar 30 2017

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[Mod[DivisorSigma[0, d[[k]]], 2] == 1] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 100}]]

a(n) = [x^n] Product_{d^2|n} 1/(1 - x^(d^2)).
a(n) = 1 if n is a squarefree.
a(n) = 2 if n is a square of prime.

A284289 Number of partitions of n into prime power divisors of n (not including 1).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 7, 1, 2, 2, 10, 1, 7, 1, 10, 2, 2, 1, 34, 2, 2, 5, 13, 1, 21, 1, 36, 2, 2, 2, 72, 1, 2, 2, 73, 1, 28, 1, 19, 13, 2, 1, 249, 2, 10, 2, 22, 1, 50, 2, 127, 2, 2, 1, 419, 1, 2, 17, 202, 2, 42, 1, 28, 2, 43, 1, 1260, 1, 2, 13, 31, 2, 49, 1, 801, 23, 2, 1, 774, 2, 2, 2, 280, 1, 608

Offset: 0

Ilya Gutkovskiy, Mar 24 2017

nonn

			a(8) = 4 because 8 has 4 divisors {1, 2, 4, 8} among which 3 are prime powers {2, 4, 8} therefore we have [8], [4, 4], [4, 2, 2] and [2, 2, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A014652, A018818, A023894, A066882, A225244, A211110, A246655.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> nops(ifactors(x)[2])=1, 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, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[PrimePowerQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 90}] (* or *)
a[0]=1; a[1]=0; a[n_] := Length@IntegerPartitions[n, All, Join @@ (#[[1]]^Range[#[[2]]] & /@ FactorInteger[n])]; a /@ Range[0, 90] (* Giovanni Resta, Mar 25 2017 *)

a(n) = [x^n] Product_{p^k|n, p prime, k >= 1} 1/(1 - x^(p^k)).
a(n) = 1 if n is a prime.
a(n) = 2 if n is a semiprime.

A284464 Number of compositions (ordered partitions) of n into squarefree divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 25, 2, 34, 19, 129, 2, 1046, 2, 742, 450, 1597, 2, 44254, 2, 27517, 3321, 29967, 2, 1872757, 571, 200390, 18560, 854850, 2, 154004511, 2, 3524578, 226020, 9262157, 51886, 3353855285, 2, 63346598, 2044895, 1255304727, 2, 185493291001, 2, 1282451595, 345852035, 2972038875, 2, 6006303471178

Offset: 0

Ilya Gutkovskiy, Mar 27 2017

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to compositions

Cf. A005117, A008683, A100346, A225244, A225245, A280194.

with(numtheory):
a:= proc(n) option remember; local b, l;
      l, b:= select(issqrfree, 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..50);   # Alois P. Heinz, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[MoebiusMu[d[[k]]]^2 x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 48}]

from sympy import divisors
from sympy.ntheory.factor_ import core
from sympy.core.cache import cacheit
@cacheit
def a(n):
    l=[x for x in divisors(n) if core(x)==x]
    @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(51)]) # Indranil Ghosh, Aug 01 2017, after Maple code

a(n) = [x^n] 1/(1 - Sum_{d|n, |mu(d)| = 1} x^d), where mu(d) is the Moebius function (A008683).

a(n) = 2 if n is a prime.

A286852 Number of partitions of n into unitary prime divisors of n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 21, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 28, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 5, 1, 2, 1, 0, 2, 42, 1, 1, 2, 43, 1, 0, 1, 2, 1, 1, 2, 49, 1, 1, 0, 2, 1, 5, 2, 2, 2, 1, 1, 10, 2, 1, 2, 2, 2

Offset: 0

Ilya Gutkovskiy, Aug 01 2017

nonn

			a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are unitary prime divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].

Antti Karttunen, Table of n, a(n) for n = 0..2309
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for related partition-counting sequences

Cf. A018818, A055231, A056169, A066874, A066882, A225244, A284289.

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[GCD[n/d[[k]], d[[k]]] == 1 && PrimeQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 95}]]

A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231
unitary_prime_factors(n) = { my(ufs = factor(A055231(n))); ufs[,1]~; };
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A286852(n) = if(n<2,1-n,partitions_into(n,vecsort(unitary_prime_factors(n), , 4))); \\ Antti Karttunen, Jul 02 2018

a(n) = [x^n] Product_{p|n, p prime, gcd(p, n/p) = 1} 1/(1 - x^p).

a(n) = 0 if n is a powerful number (A001694).

A300585 Number of partitions of n into squarefree parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 2, 1, 7, 1, 12, 3, 4, 8, 29, 2, 42, 8, 18, 18, 87, 7, 71, 36, 67, 32, 234, 3, 319, 98, 126, 118, 192, 31, 772, 205, 293, 98, 1347, 21, 1763, 338, 295, 574, 2973, 116, 2290, 298, 1359, 932, 6287, 214, 2670, 843, 2744, 2334, 12828, 66, 16155, 3620, 2835, 4584, 8380

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

			a(14) = 3 because we have [11, 3], [6, 5, 3] and [5, 3, 3, 3].

Index entries for sequences related to partitions

Cf. A005117, A073576, A098743, A128515, A225244, A300580, A300586.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] != 0 && SquareFreeQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 65}]

A327641 Number of partitions of n into divisors d of n such that n/d is squarefree.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 8, 2, 2, 2, 11, 2, 8, 2, 14, 14, 2, 2, 8, 2, 11, 18, 20, 2, 8, 2, 23, 2, 14, 2, 742, 2, 2, 26, 29, 26, 8, 2, 32, 30, 11, 2, 1654, 2, 20, 14, 38, 2, 8, 2, 11, 38, 23, 2, 8, 38, 14, 42, 47, 2, 742, 2, 50, 18, 2, 44, 5257, 2, 29, 50, 5066, 2, 8, 2, 59, 14

Offset: 0

Ilya Gutkovskiy, Sep 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A008683, A018818, A225244, A225245, A246655 (positions of 2's).

[1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)|IsSquarefree(n div d)}):n in [1..75]]; // Marius A. Burtea, Sep 20 2019

a[n_] := SeriesCoefficient[Product[1/(1 - MoebiusMu[n/d]^2 x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 75}]

A327641(n) = if(!n, 1, my(p = Ser(1, 'x, 1+n)); fordiv(n, d, if(issquarefree(n/d), p /= (1 - 'x^d))); polcoef(p, n)); \\ Antti Karttunen, Jan 28 2025

a(n) = [x^n] Product_{d|n} 1 / (1 - mu(n/d)^2 * x^d).

A378843 Number of compositions (ordered partitions) of n into distinct squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 7, 1, 0, 0, 1, 1, 24, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 151, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 31, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 864, 1, 1, 0, 0, 1, 127, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 7, 1, 0

Offset: 0

Ilya Gutkovskiy, Dec 09 2024

nonn

From Robert Israel, Dec 15 2024: (Start)
If n is squarefree, a(n) >= 1, as [n] is a composition.
If n = b * c where b and c are coprime and c is squarefree, then a(n) >= a(b), as for any composition C of b into distinct squarefree divisors, multiplying each element of C by c gives a composition of n into distinct squarefree divisors. (End)

			a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].
a(12) = 24 because we have [6, 3, 2, 1] and 4! = 24 permutations.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A005117, A087188, A225244, A225245, A284464, A331846, A331927.

ptns:= proc(S,n) option remember;
  # subsets of S with sum n
  local m,s;
  if convert(S,`+`) < n then return {} fi;
  if n = 0 then return {{}} fi;
  s:= max(S);
  if s > n then return procname(select(`<=`,S,n),n) fi;
  map(t -> t union {s}, procname(S minus {s},n-s)) union procname(S minus {s}, n)
  end proc:
sfd:= proc(n) map(convert,combinat:-powerset(numtheory:-factorset(n)),`*`) end proc:
f:= proc(n) local t;
     add((nops(t))!, t = ptns(sfd(n),n))
end proc:
map(f, [$0..100]); # Robert Israel, Dec 15 2024

a[n_] := Module[{d = Select[Divisors[n], SquareFreeQ]}, Total[(Length /@ Select[Subsets[d], Total[#] == n &])!]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Dec 10 2024 *)

cp's OEIS Frontend

A018818 Number of partitions of n into divisors of n.

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A073576 Number of partitions of n into squarefree parts.

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A225245 Number of partitions of n into distinct squarefree divisors of n.

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A284345 Number of partitions of n into squares dividing n.

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A284289 Number of partitions of n into prime power divisors of n (not including 1).

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A284464 Number of compositions (ordered partitions) of n into squarefree divisors of n.

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A286852 Number of partitions of n into unitary prime divisors of n.

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