A116540 -id:A116540 - cp's OEIS frontend

A283877 Number of non-isomorphic set-systems of weight n.

1, 1, 2, 4, 9, 18, 44, 98, 244, 605, 1595, 4273, 12048, 34790, 104480, 322954, 1031556, 3389413, 11464454, 39820812, 141962355, 518663683, 1940341269, 7424565391, 29033121685, 115921101414, 472219204088, 1961177127371, 8298334192288, 35751364047676, 156736154469354

Offset: 0

Gus Wiseman, Mar 17 2017

nonn

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.

			Non-isomorphic representatives of the a(4)=9 set-systems are:
((1234)),
((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
((1)(2)(3)(4)).

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

Euler transform of A300913.

a(0) = 1 prepended and terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A276625 Finitary numbers. Matula-Goebel numbers of rooted identity trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 39, 41, 47, 55, 58, 62, 65, 66, 78, 79, 82, 87, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 143, 145, 155, 158, 165, 167, 174, 179, 186, 195, 202, 205, 211, 218, 226, 235, 237, 246, 254, 257, 271, 274, 282, 286, 290, 293, 303, 310, 313, 317, 319, 327, 330

Offset: 1

Gus Wiseman, Sep 29 2016

nonn

For any positive integer n the following are equivalent:
(1) n is a finitary number.
(2) prime(n) is a finitary number.
(3) n is a product of distinct finitary prime numbers.
These conditions are necessary and sufficient to define an infinite set of positive integers but do not specify how this set should be enumerated or indexed (is there a more natural way? viz. A215366) so here they are listed in increasing order of the corresponding Matula-Goebel numbers. The following comment is from A007097.
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012
Notes on use of the word "finitary": It is possible to have a finite set containing an infinite set. For example {{1,2,3...}} contains only one element. In contrast, a finitary set is a finite set whose elements are also required to be finitary sets. There are also no multisets allowed in finitary sets, although you can have repeated elements. For example {{{}},{{},{{}}}} is still considered a finitary set even though the multiset union {{},{},{{}}} is not a set. The finitary numbers of A276625 refer to multisets (trees) that don't involve any proper multisets (i.e. only sets). This is in addition to the (somewhat redundant) meaning of finitary sets as described in this comment on A004111 "There is a natural correspondence between rooted identity trees and finitary sets (sets whose transitive closure is finite); each node represents a set, with the children of that node representing the members of that set. When the set corresponding to an identity tree is written out using braces, there is one set of braces for each node of the tree; thus a(n) is also the number of sets that can be made using n pairs of braces. - Franklin T. Adams-Watters, Oct 25 2011" - Gus Wiseman, Oct 03 2016

			This sequence is proposed to be a canonical representation for rooted identity trees. The first thirty terms are the following.
1  ()           26 (()(()(())))     62  (()((((())))))
2  (())         29 ((()((()))))     65  (((()))(()(())))
3  ((()))       30 (()(())((())))   66  (()(())(((()))))
5  (((())))     31 (((((())))))     78  (()(())(()(())))
6  (()(()))     33 ((())(((()))))   79  ((()(((())))))
10 (()((())))   39 ((())(()(())))   82  (()((()(()))))
11 ((((()))))   41 (((()(()))))     87  ((())(()((()))))
13 ((()(())))   47 (((())((()))))   93  ((())((((())))))
15 ((())((()))) 55 (((()))(((())))) 94  (()((())((()))))
22 (()(((())))) 58 (()(()((()))))   101 ((()(()(()))))
We build the sequence as follows: The empty product is 1, so by (3) 1 is finitary. So is prime(1) = 2 by (2), so is prime(2) = 3 by (2), so is prime(3) = 5 by (2), so is 2*3 = 6 by (3), and so on. - _N. J. A. Sloane_, Oct 03 2016

Index entries for sequences related to Matula-Goebel numbers

Cf. A000040 (prime numbers), A000720 (PrimePi).
Cf. A004111 (identity trees), A116540 (set multipartitions). Contained in A005117 (squarefree numbers). Contains A076146 (ordinal numbers), A007097 (rooted paths), A277098 (finitary primes).
Cf. A206497 (automorphism group sizes), A348066 (reduce to identity tree).

primeMS[n_Integer?Positive]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_Integer?Positive]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
fin[n_Integer?Positive]:=If[n===1,1,Block[{x=fin[n-1]+1},While[Not[finitaryQ[x]],x++];x]];
Array[fin,200]

a(n) = primePi(A277098(n)).

A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

Original entry on oeis.org

1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664

Offset: 1

Peter J. Cameron

Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.
The EULERi transform (A056156) is also interesting.
a(n) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n. - Gus Wiseman, Mar 17 2017

			E.g. a(2) = 3: two ones in same row, two ones in same column, or neither.
a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9.
There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation:
  [1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1]
  [0 1 0] [0 0 1] [1 0] [1 0] ....... [1].
  [0 0 1] ....... [0 1] ............. [1]
Non-isomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)). - _Gus Wiseman_, Mar 17 2017

Aliaksandr Siarhei, Table of n, a(n) for n = 1..102
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
Peter J. Cameron, Problems on Permutation Groups, see Problem 3
Index entries for sequences related to binary matrices.

Main diagonal of A321609.
Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.
Cf. A057149, A057150, A057151, A057152.
Cf. A034691, A056156, A089259, A116540, A283877.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.

More terms and formula from Vladeta Jovovic, Jul 29 2000
a(19)-a(28) from Max Alekseyev, Jul 22 2009
a(29)-a(102) from Aliaksandr Siarhei, Dec 13 2013
Name edited by Gus Wiseman, Dec 18 2018

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756

Offset: 0

N. J. A. Sloane, Dec 23 2003

nonn

Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008

Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016

			From _Gus Wiseman_, Oct 22 2018: (Start)
The a(6) = 22 set multipartitions of integer partitions of 6:
  (6)  (15)    (123)      (12)(12)      (1)(1)(1)(12)    (1)(1)(1)(1)(1)(1)
       (24)    (1)(14)    (1)(1)(13)    (1)(1)(1)(1)(2)
       (1)(5)  (1)(23)    (1)(2)(12)
       (2)(4)  (2)(13)    (1)(1)(1)(3)
       (3)(3)  (3)(12)    (1)(1)(2)(2)
               (1)(1)(4)
               (1)(2)(3)
               (2)(2)(2)
(End)

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

Row sums of A285229 and of A360763.

Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.

with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008
# second Maple program:
with(numtheory):
b:= proc(n, i)
      if n<0 or n>i*(i+1)/2 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n,i):= b(n-i, i-1) +b(n, i-1)
      fi
    end:
a:= proc(n) option remember; `if` (n=0, 1,
       add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 11 2011

max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)
b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018

A050320 Number of ways n is a product of squarefree numbers > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 1

Offset: 1

Christian G. Bower, Sep 15 1999

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
Broughan shows (Theorem 8) that the average value of a(n) is k exp(2*sqrt(log n)/sqrt(zeta(2)))/log(n)^(3/4) where k is about 0.18504.  - Charles R Greathouse IV, May 21 2013
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime indices of n. For example, the a(n) set multipartitions for n = 2, 6, 36, 60, 360 are:
  {1}  {12}    {12}{12}      {1}{123}      {1}{12}{123}
       {1}{2}  {1}{2}{12}    {12}{13}      {12}{12}{13}
               {1}{1}{2}{2}  {1}{1}{23}    {1}{1}{12}{23}
                             {1}{2}{13}    {1}{1}{2}{123}
                             {1}{3}{12}    {1}{2}{12}{13}
                             {1}{1}{2}{3}  {1}{3}{12}{12}
                                           {1}{1}{1}{2}{23}
                                           {1}{1}{2}{2}{13}
                                           {1}{1}{2}{3}{12}
                                           {1}{1}{1}{2}{2}{3}
(End)

			For n = 36 we have three choices as 36 = 2*2*3*3 = 6*6 = 2*3*6 (but no factorizations with factors 4, 9, 12, 18 or 36 are allowed), thus a(36) = 3. - _Antti Karttunen_, Oct 21 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000
Kevin Broughan, Quadrafree factorisatio numerorum, Rocky Mountain J. Math. 44 (3) (2014) 791-807.
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A005117, A050325. a(p^k)=1. a(A002110)=A000110.
a(n!)=A103774(n).
Cf. A206778.
Differs from A259936 for the first time at n=36.
A050326 is the strict case.
A045778 counts strict factorizations.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
Cf. A008480, A124010, A317829, A318360.

a050320 n = h n $ tail $ a206778_row n where
   h 1 _          = 1
   h _ []         = 0
   h m fs'@(f:fs) =
     if f > m then 0 else if r > 0 then h m fs else h m' fs' + h m fs
     where (m', r) = divMod m f
-- Reinhard Zumkeller, Dec 16 2013

sub[w_, e_] := Block[{v = w}, v[[e]]--; v]; ric[w_, k_] := If[Max[w] == 0, 1, Block[{e, s, p = Flatten@Position[Sign@w, 1]}, s = Select[Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; sig[w_] := sig[w] = ric[w, 1];  a[n_] := sig@ Sort[Last /@ FactorInteger[n]]; Array[a, 103] (* Giovanni Resta, May 21 2013 *)
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Aug 20 2020 *)

Dirichlet g.f.: Product_{n is squarefree and > 1} (1/(1-1/n^s)).
a(n) = A050325(A101296(n)). - R. J. Mathar, May 26 2017
a(n!) = A103774(n); a(A006939(n)) = A337072(n). - Gus Wiseman, Aug 20 2020

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A364348, A364537, A350845 in not containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  102: {1,2,7}
  105: {2,3,4}
  108: {1,1,2,2,2}

Partitions of this type are counted by A351293, complement A239455.
The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],!MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 5, 3, 2, 1, 6, 1, 2, 3, 15, 1, 9, 1, 6, 3, 2, 1, 21, 4, 2, 16, 6, 1, 10, 1, 52, 3, 2, 4, 35, 1, 2, 3, 22, 1, 10, 1, 6, 19, 2, 1, 83, 5, 13, 3, 6, 1, 66, 4, 22, 3, 2, 1, 41, 1, 2, 20, 203, 4, 10, 1, 6, 3, 14, 1, 153, 1, 2, 26, 6, 5, 10, 1

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(12) = 6 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A001055, A007716, A049311, A116540, A181821, A188392, A255906.

Cf. A318283, A318284, A318286, A318361, A318362, A318369.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,80}]

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig), s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + O(x*x^n))); s+=prod(i=1, #sig, polcoef(q,sig[i]))*permcount(p)); s/n!}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s<=2, if(#s==1, 1, min(s[1],s[2])+1), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018

a(n) = A050320(A181821(n)).
From Andrew Howroyd, Dec 10 2018:(Start)
a(p) = 1 for prime(p).
a(prime(i)*prime(j)) = min(i,j) + 1.
a(prime(n)^k) = A188392(n,k). (End)

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 3, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 3, 3, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 3, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A357982 at a(25) = 3, A357982(25) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.

For constant instead of strict partitions see A381453, A355733, A381455, A000688.
Positions of 1 are A003586.
The upper version is A381078, before sums A050320.
For distinct block-sums see A381634, A381633, A381806.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on set systems: A050342, A116539, A296120, A318361.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[n]]]],{n,100}]

a(A002110(n)) = A381808(n).

A381633 Number of ways to partition the prime indices of n into sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 5, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 09 2025

nonn

First differs from A050326 at 30, 60, 70, 90, ...
First differs from A339742 at 42, 66, 78, 84, ...
First differs from A381634 at a(210) = 12, A381634(210) = 10.
Also the number of factorizations on n into squarefree numbers > 1 with distinct sums of prime indices.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The A050320(60) = 6 ways to partition {1,1,2,3} into sets are:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
Of these, only the following have distinct block-sums:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
So a(60) = 3.

Without distinct block-sums we have A050320, after sums A381078 (lower A381454).
For distinct blocks instead of sums we have A050326, after sums A381441, see A358914.
Taking block-sums (and sorting) gives A381634.
For constant instead of strict blocks we have A381635, see A381716, A381636.
Positions of 0 are A381806, superset of A293243.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A001055 count multiset partitions of prime indices, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A213242, A299202, A300385, A317142, A317143, A321469.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[sfacs[n],UnsameQ@@hwt/@#&]],{n,100}]

cp's OEIS Frontend

A283877 Number of non-isomorphic set-systems of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A276625 Finitary numbers. Matula-Goebel numbers of rooted identity trees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A089259 Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A050320 Number of ways n is a product of squarefree numbers > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Views