A047968 -id:A047968 - cp's OEIS frontend

A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions.

1, 1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 100, 119, 167, 209, 296, 347, 489, 582, 775, 945, 1254, 1481, 1951, 2334, 2980, 3580, 4564, 5386, 6841, 8118, 10085, 12012, 14862, 17526, 21636, 25524, 31082, 36694, 44582, 52255, 63260, 74170, 88931, 104302

Offset: 0

N. J. A. Sloane

Starting (1, 1, 2, 3, 6, 7, 14, ...), = row sums of triangle A137585. - Gary W. Adamson, Jan 27 2008
Triangle A168532 has aerated variants of this sequence in each column starting with offset 1, row sums = A000041. -  Gary W. Adamson, Nov 28 2009
A partition is aperiodic iff its multiplicities are relatively prime, i.e., its Heinz number (A215366) is not a perfect power (A007916). - Gus Wiseman, Dec 19 2017
This sequence is monotonically increasing; each partition of n-1 can have a part of size 1 added to it to get a partition counted in a(n). - Franklin T. Adams-Watters, Jul 24 2020

			Of the 11 partitions of 6, we must exclude 6, 4+2, 3+3 and 2+2+2, so a(6) = 11 - 4 = 7.
For n=6, 2+2+1+1 is periodic because it can be written 2*(2+1), similarly 1+1+1+1+1+1, 3+3 and 2+2+2.
The a(6) = 7 partitions into relatively prime parts are (51), (411), (321), (3111), (2211), (21111), (111111). The a(6) = 7 aperiodic partitions are (6), (51), (42), (411), (321), (3111), (21111). - _Gus Wiseman_, Dec 19 2017

H. W. Gould, personal communication.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Mohamed El Bachraoui, On the Parity of p(n,3) and p_psi(n,3), Contributions to Discrete Mathematics, Vol. 5.2 (2010).
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Mircea Merca and Maxie D. Schmidt, Generating Special Arithmetic Functions by Lambert Series Factorizations, arXiv:1706.00393 [math.NT], 2017. See Remark 3.4.
N. J. A. Sloane, Transforms

Cf. A000041, A018783.

Cf. A000740, A007916, A047968, A055892, A100953, A137585, A168532, A281116.

p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]]; g[n_, j_] := Apply[GCD, Part[p[n], j]]; h[n_] := Table[g[n, j], {j, 1, l[n]}]; Join[{1}, Table[Count[h[n], 1], {n, 1, 20}]]
(* Clark Kimberling, Mar 09 2012 *)
a[0] = 1; a[n_] := Sum[ MoebiusMu[n/d] * PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 03 2013 *)

N=66; x='x+O('x^N); gf=2+sum(n=1,N, (1/eta(x^n))*moebius(n)); Vec(gf) \\ Joerg Arndt, May 11 2013

print1("1, "); for(n=1,46,my(s=0);forpart(X=n,s+=gcd(X)==1);print1(s,", ")) \\ Hugo Pfoertner, Mar 27 2020

from sympy import npartitions, mobius, divisors
def a(n): return 1 if n==0 else sum(mobius(n//d)*npartitions(d) for d in divisors(n)) # Indranil Ghosh, Apr 26 2017

Möbius transform of A000041. - Christian G. Bower, Jun 11 2000
Product_{n>0} 1/(1-q^n) = 1 + Sum_{n>0} a(n)*q^n/(1-q^n). - Mamuka Jibladze, Nov 14 2015
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019
a(n) <= p(n) <= a(n+1), where p(n) is the number of partitions of n (A000041). - Franklin T. Adams-Watters, Jul 24 2020

Corrected and extended by David W. Wilson, Aug 15 1996

Additional name from Christian G. Bower, Jun 11 2000

A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

Original entry on oeis.org

1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758

Offset: 0

Wouter Meeussen, Aug 21 2001

These are different from plane partitions.
For ordered partitions of partitions see A055887 which may be computed from A036036 and A048996. - Alford Arnold, May 19 2006
Twice partitioned numbers correspond to triangles (or compositions) in the multiorder of integer partitions. - Gus Wiseman, Oct 28 2015

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n=1..500 from T. D. Noe)
Vaclav Kotesovec, Screenshot - A closed form formula for the constant c
Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Sequences enumerating triangles of integer partitions
Gus Wiseman, On the Categorical Structure of Weakly Ordered Sequences of Integer Partitions

Cf. A063835, A196545.
Cf. A036036, A048996, A055887.
Cf. A006906, A270995.
Cf. A007425, A047966, A047968, A271619, A279375, A279784-A279791.
The strict case is A296122.
Row sums of A321449.
Column k=2 of A323718.
Without singletons we have A327769, A358828, A358829.
For odd lengths we have A358823, A358824.
For distinct lengths we have A358830, A358912.
For strict partitions see A358914, A382524.
A000041 counts integer partitions, strict A000009.
A001970 counts multiset partitions of integer partitions.
Cf. A075900, A358831, A358833, A358906, A358908, A358909.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 26 2015

Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */

G.f.: 1/Product_{k>0} (1-A000041(k)*x^k). n*a(n) = Sum_{k=1..n} b(k)*a(n-k), a(0) = 1, where b(k) = Sum_{d|k} d*A000041(d)^(k/d) = 1, 5, 10, 29, 36, 110, 106, ... . - Vladeta Jovovic, Jun 19 2003
From Vaclav Kotesovec, Mar 27 2016: (Start)
a(n) ~ c * 5^(n/4), where
c = 96146522937.7161898848278970039269600938032826... if n mod 4 = 0
c = 96146521894.9433858914667933636782092683849082... if n mod 4 = 1
c = 96146522937.2138934755566928890704687838407524... if n mod 4 = 2
c = 96146521894.8218716328341714149619262713426755... if n mod 4 = 3
(End)

a(0)=1 prepended by Alois P. Heinz, Nov 26 2015

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

Original entry on oeis.org

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004
Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018
Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  Gary W. Adamson, Jun 08 2009

			The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)
Table[DivisorSum[n,PartitionsQ],{n,20}] (* Gus Wiseman, Apr 16 2018 *)

N = 66; q='q+O('q^N);
D(q)=eta(q^2)/eta(q); \\ A000009
Vec( sum(e=1,N,D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

Original entry on oeis.org

1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145, 524827, 1048645, 2098179, 4194305, 8390831, 16777233, 33558531, 67109125, 134225995, 268435457, 536887863, 1073741825, 2147516555, 4294968325, 8590000131

Offset: 1

Erich Friedman

nonn

Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - Gary W. Adamson, Aug 14 2008
Number of constant multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The a(4) = 11 constant multiset partitions:
  (1)(1)(1)(1)
    (11)(11)
    (12)(12)
     (1111)
     (1222)
     (1122)
     (1112)
     (1233)
     (1223)
     (1123)
     (1234)
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.
Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.
Cf. A289508.
Sums of the form Sum_{d|n} q^(d-1): this sequence (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
[A034729(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014

Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 08 2014 *)

A034729(n) = sumdiv(n,k,2^(k-1)) \\ Michael B. Porter, Mar 11 2010

{a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

{a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,1/(1 - x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

from sympy import divisors
def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 15 2022

def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
[A034729(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

G.f.: Sum_{n>0} x^n/(1-2*x^n). - Vladeta Jovovic, Nov 14 2002
a(n) = 1/2 * A055895(n). - Joerg Arndt, Aug 14 2012
G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - Paul D. Hanna, Aug 21 2014
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - Paul D. Hanna, Aug 21 2014
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 09 2014
a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - Gus Wiseman, Sep 16 2018
a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - Gus Wiseman, Sep 16 2018

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

			From _Gus Wiseman_, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
Row sums of A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k],{k,1,nmax}],{x,0,nmax}],x]

A319066 Number of partitions of integer partitions of n where all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 59, 82, 128, 176, 273, 371, 553, 768, 1119, 1544, 2235, 3084, 4410, 6111, 8649, 11982, 16901, 23383, 32780, 45396, 63365, 87622, 121946, 168407, 233605, 322269, 445723, 613922, 847131, 1164819, 1603431, 2201370, 3023660, 4144124, 5680816

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{1,1},{1,1}}      {{1,1,1,2}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001970, A047968, A261049, A279787, A305551, A306017, A319056.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Length/@#&]],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k*y + O(x*x^n))); concat([1], sum(k=1, n, EulerT(Vec(polcoef(p, k, y), -n))))} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A035310 Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

Original entry on oeis.org

1, 4, 12, 47, 170, 750, 3255, 16010, 81199, 448156, 2579626, 15913058, 102488024, 698976419, 4976098729, 37195337408, 289517846210, 2352125666883, 19841666995265, 173888579505200, 1577888354510786, 14820132616197925, 143746389756336173, 1438846957477988926

Offset: 1

Alford Arnold

Ways of partitioning an n-multiset with multiplicities some partition of n.

Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019

			a(3) = 12 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=3, f(12)=4, f(30)=5 and 3+4+5 = 12.
From _Gus Wiseman_, Dec 31 2019: (Start)
The a(1) = 1 through a(3) = 12 multiset partitions of strongly normal multisets:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A025487, A000041, A000110, A035098, A080688.
Sequence A035341 counts the ordered cases. Tables A093936 and A095705 distribute the values; e.g. 81199 = 30 + 536 + 3036 + 6181 + 10726 + 11913 + 14548 + 13082 + 21147.
Cf. A035341, A093936, A095705.
Row sums of A317449.
The uniform case is A317584.
The case with empty intersection is A317755.
The strict case is A317775.
The constant case is A047968.
The set-system case is A318402.
The case of strict parts is A330783.
Multiset partitions of integer partitions are A001970.
Unlabeled multiset partitions are A007716.
Cf. A001055, A255906, A269134, A316652, A317654, A318284, A330467, A330471, A330475, A330675.

with(numtheory):
g:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
b:= proc(n, i, l)
      `if`(n=0, g(mul(ithprime(t)^l[t], t=1..nops(l))$2),
      `if`(i<1, 0, add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=1..10);  # Alois P. Heinz, May 26 2013

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; b[n_, i_, l_] := If[n == 0, g[p = Product[Prime[t]^l[[t]], {t, 1, Length[l]}], p], If[i < 1, 0, Sum[b[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := b[n, n, {}]; Table[Print[an = a[n]]; an, {n, 1, 13}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

from sympy.core.cache import cacheit
from sympy import divisors, isprime, prime
from operator import mul
@cacheit
def g(n, k):
    return (0 if n > k else 1) + (0 if isprime(n) else sum(g(n//d, d) for d in divisors(n)[1:-1] if d <= k))
@cacheit
def b(n, i, l):
    if n==0:
        p = reduce(mul, (prime(t + 1)**l[t] for t in range(len(l))))
        return g(p, p)
    else:
        return 0 if i<1 else sum([b(n - i*j, i - 1, l + [i]*j) for j in range(n//i + 1)])
def a(n):
    return b(n, n, [])
for n in range(1, 11): print(a(n)) # Indranil Ghosh, Aug 19 2017, after Maple code

More terms from Erich Friedman.
81199 from Alford Arnold, Mar 04 2008
a(10) from Alford Arnold, Mar 31 2008
a(10) corrected by Alford Arnold, Aug 07 2008
a(11)-a(13) from Alois P. Heinz, May 26 2013
a(14) from Alois P. Heinz, Sep 27 2014
a(15) from Alois P. Heinz, Jan 10 2015
Terms a(16) and beyond from Andrew Howroyd, Dec 30 2020

A296122 Number of twice-partitions of n with no repeated partitions.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 40, 77, 157, 285, 552, 1018, 1921, 3484, 6436, 11622, 21082, 37550, 67681, 119318, 211792, 372003, 653496, 1137185, 1986234, 3429650, 5935970, 10205907, 17537684, 29958671, 51189932, 86967755, 147759421, 249850696, 422123392, 710495901

Offset: 0

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.

			The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000009, A000041, A047968, A063834, A261049, A273873, A279375, A295279, A296121.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2017

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p],UnsameQ@@#&],{p,IntegerPartitions[n]}]],{n,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*
     Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(15)-a(34) from Robert G. Wilson v, Dec 06 2017

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 3, 9, 9, 12, 12, 20, 18, 28, 27, 37, 42, 55, 51, 74, 80, 98, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of strict integer partitions of n with no part dividing all the others.

			The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
  0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
                         52        72    73    74    543   85    95     96
                                   432   532   83    732   94    A4     B4
                                               92          A3    B3     D2
                                               542         B2    653    654
                                               632         643   743    753
                                                           652   752    762
                                                           742   932    843
                                                           832   5432   852
                                                                        942
                                                                        A32
                                                                        6432

The complement is counted by A097986 (non-strict: A083710, rank: A339563).
The complement with no 1's is A098965 (non-strict: A083711).
The non-strict version is A338470.
The Heinz numbers of these partitions are A339562 (non-strict: A342193).
The case with greatest part not divisible by all others is A343379.
The case with greatest part divisible by all others is A343380.
A000009 counts strict partitions (non-strict: A000041).
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]

a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Gus Wiseman, Mar 23 2021

nonn

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

cp's OEIS Frontend

A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions.

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A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

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A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

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A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

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A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

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A319066 Number of partitions of integer partitions of n where all parts have the same length.

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