A379957 -id:A379957 - 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

A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 10, 1, 8, 4, 10, 1, 15, 1, 17, 5, 16, 1, 26, 2, 22, 5, 29, 1, 37, 1, 36, 7, 38, 4, 57, 1, 48, 9, 65, 1, 73, 1, 77, 13, 76, 1, 108, 2, 99, 11, 117, 1, 130, 5, 145, 14, 142, 1, 189, 1, 168, 19, 202, 5, 223, 1, 241, 17, 247, 1, 309, 1, 286, 24, 333, 4

Offset: 0

Henry Bottomley, Jul 05 2002

nonn

First differs from A322968 at a(12) = 10, A322968(12) = 9.

			a(5)=1 since the only partition without 1 as a part is 5 (a power of 5). a(6)=4 since 6 can be written as 6 (powers of 6), 3+3 (powers of 3) and 4+2 and 2+2+2 (both powers of 2).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (C)
            (22)       (33)        (44)    (333)  (55)          (66)
                       (42)        (422)          (82)          (84)
                       (222)       (2222)         (442)         (93)
                                                  (4222)        (444)
                                                  (22222)       (822)
                                                                (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
(End)
Compare above to the example section of A379957. - _Antti Karttunen_, Jan 23 2025

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions.

Cf. A018819, A023894, A052410, A072720, A072721, A102430, A175082, A322900, A322902, A322903, A322968.

Cf. also A379957.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],SameQ@@radbase/@#]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)

a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&!ispower(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025

seq(n)={Vec(1 + sum(d=2, n, if(!ispower(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025

a(n) = A072721(n)-A072721(n-1). a(p)=1 for p prime.
a(n) = A322900(n) - 1. - Gus Wiseman, Jan 01 2019
G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A175082(k)^j)). - Andrew Howroyd, Jan 23 2025
For n >= 1, a(n) >= A379957(n). - Antti Karttunen, Jan 23 2025

A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 9, 1, 8, 4, 10, 1, 14, 1, 16, 5, 16, 1, 24, 2, 22, 5, 28, 1, 37, 1, 36, 7, 38, 4, 55, 1, 48, 9, 63, 1, 73, 1, 76, 12, 76, 1, 105, 2, 98, 11, 116, 1, 128, 5, 143, 14, 142, 1, 186, 1, 168, 18, 202, 5, 223, 1, 240, 17, 247, 1, 305, 1, 286, 23

Offset: 0

Gus Wiseman, Jan 01 2019

nonn

First differs from A072721 at a(12) = 9, A072721(12) = 10.

First differs from A379957 at a(16) = 10, A379957(16) = 9.

			The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (66)
            (22)       (33)        (44)    (333)  (55)          (84)
                       (42)        (422)          (82)          (93)
                       (222)       (2222)         (442)         (444)
                                                  (4222)        (822)
                                                  (22222)       (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
The a(20) = 16 integer partitions:
  (10,10), (16,4),
  (8,8,4), (16,2,2),
  (5,5,5,5), (8,4,4,4), (8,8,2,2),
  (4,4,4,4,4), (8,4,4,2,2),
  (4,4,4,4,2,2), (8,4,2,2,2,2),
  (4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2),
  (4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2).

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions.

Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912, A379957.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],And@@powsqfQ/@#,SameQ@@radbase/@#]&]],{n,30}]

a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&issquarefree(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025

seq(n)={Vec(1 + sum(d=2, n, if(issquarefree(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025

From  Andrew Howroyd, Jan 23 2025: (Start)
G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A005117(k)^j)).
a(p) = 1 for prime p. (End)

a(66) onwards from Andrew Howroyd, Jan 23 2025

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A018818 Number of partitions of n into divisors of n.

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A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).

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A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

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