A002846 -id:A002846 - cp's OEIS frontend

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A000688 at a(144) = 9, A000688(144) = 10.
First differs from A295879 at a(128) = 15, A295879(128) = 13.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into prime powers > 1.
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).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
  {{1,1},{2,2}}
  {{1},{1},{2,2}}
  {{2},{2},{1,1}}
  {{1},{1},{2},{2}}
with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
  {{2,2},{1,1,1,1}}
  {{1},{2,2},{1,1,1}}
  {{2},{2},{1,1,1,1}}
  {{1,1},{1,1},{2,2}}
  {{1},{1},{1,1},{2,2}}
  {{1},{2},{2},{1,1,1}}
  {{2},{2},{1,1},{1,1}}
  {{1},{1},{1},{1},{2,2}}
  {{1},{1},{2},{2},{1,1}}
  {{1},{1},{1},{1},{2},{2}}
with block-sums: {4,4}, {1,3,4}, {2,2,4}, {2,2,4}, {1,1,2,4}, {1,2,2,3}, {2,2,2,2}, {1,1,1,1,4}, {1,1,2,2,2}, {1,1,1,1,2,2}, of which 9 are distinct, so a(144) = 9.
The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
  (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
  (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
        (111)  (31)    (41)     (411)   (42)      (421)    (52)
               (211)   (221)    (2211)  (51)      (2221)   (61)
               (1111)  (311)            (222)     (4111)   (322)
                       (2111)           (321)     (22111)  (331)
                       (11111)          (411)              (421)
                                        (2211)             (511)
                                        (3111)             (2221)
                                        (21111)            (3211)
                                        (111111)           (4111)
                                                           (22111)
                                                           (31111)
                                                           (211111)
                                                           (1111111)

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A000688.
Positions of 1 are A005117.
There is a chain from the prime indices of n to a singleton iff n belongs to A300273.
The lower version is A381453.
For distinct blocks we have A381715, before sum A050361.
For distinct block-sums we have A381716, before sums A381635 (zeros A381636).
Other multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326) see A381441 (upper).
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633) see A381634, A293243.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
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.
Cf. A000720, A001222, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.

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

a(s) = 1 for any squarefree number s.

a(p^k) = A000041(k) for any prime p.

A300385 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 11, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 19, 1, 1, 2, 2, 1, 3, 1, 14, 2, 1, 1, 10, 1, 1, 1, 5, 1, 10, 1, 2, 1, 1, 1, 33, 1, 2, 2, 7, 1, 3, 1, 5, 3

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(36) = 6 maximal chains are the rows:
(2211)<(222)<(42)<(6)
(2211)<(411)<(42)<(6)
(2211)<(411)<(51)<(6)
(2211)<(321)<(42)<(6)
(2211)<(321)<(51)<(6)
(2211)<(321)<(33)<(6)

Antti Karttunen, Table of n, a(n) for n = 1..12288
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300384.

chc[ptn_]:=If[Length[ptn]===1,1,Total[chc/@Union[ReplaceList[ptn,{a___,x_,b___,y_,c___}:>Sort[{x+y,a,b,c},Greater]]]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[chc[Reverse[primeMS[n]]],{n,100}]

A300385(n) = if(1==n,0,if(bigomega(n)<=2,1,my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A300385((n/(f[i,1]*f[j,1])*prime(primepi(f[i,1])+primepi(f[j,1])))))); (s))); \\ Antti Karttunen, Oct 06 2018

memoA300385 = Map();
A300385(n) = if(1==n,0,if(bigomega(n)<=2,1,if(mapisdefined(memoA300385,n),mapget(memoA300385,n),my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A300385(prime(primepi(f[i,1])+primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA300385,n,s); (s)))); \\ (A memoized implementation). - Antti Karttunen, Oct 07 2018

a(1) = 0; for n > 1, if A001222(n) <= 2 [when n is a prime or semiprime], a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)+primepi(q)) * (n/(p*q))), where p ranges over all distinct primes dividing n, and q also ranges over primes dividing n, but with condition that q > p if p is a unitary prime factor of n, otherwise q >= p. Here primepi = A000720. - Antti Karttunen, Oct 07 2018

More terms from Antti Karttunen, Oct 06 2018

A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 33, 21, 8, 1, 11, 91, 104, 36, 10, 1, 15, 298, 452, 238, 55, 12, 1, 22, 910, 2335, 1430, 455, 78, 14, 1, 30, 3017, 11992, 10179, 3505, 775, 105, 16, 1, 42, 9945, 66810, 74299, 31881, 7297, 1218, 136, 18, 1, 56

Offset: 1

Gus Wiseman, Jan 27 2019

			Array begins:
      k=1:  k=2:  k=3:  k=4:  k=5:  k=6:
  n=1:  1     1     1     1     1     1
  n=2:  2     4     6     8    10    12
  n=3:  3    10    21    36    55    78
  n=4:  5    33   104   238   455   775
  n=5:  7    91   452  1430  3505  7297
  n=6: 11   298  2335 10179 31881 80897
Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:
  {{111}}          {{112}}          {{123}}
  {{1}{11}}        {{1}{12}}        {{1}{23}}
  {{1}}{{11}}      {{2}{11}}        {{1}}{{23}}
  {{1}{1}{1}}      {{1}}{{12}}      {{1}{2}{3}}
  {{1}}{{1}{1}}    {{1}{1}{2}}      {{1}}{{2}{3}}
  {{1}}{{1}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}
                   {{1}}{{1}{2}}
                   {{2}}{{1}{1}}
                   {{1}}{{1}}{{2}}

Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).
Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).
Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

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]]]];
undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Apply[Rule,Table[{undats[m][[i]],i},{i,Length[undats[m]]}],{1}]],First[Sort[expnorm[m,1]]]]];
expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#1>=aft&]}]},Union@@(expnorm[#1,aft+1]&)/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]]]];
strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];
kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];
Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]

a(46)-a(56) from Robert Price, May 11 2021

A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

Original entry on oeis.org

1, 1, 2, 6, 20, 90, 468, 2910, 20644, 165874, 1484344, 14653890, 158136988, 1852077284, 23394406084, 317018563806, 4587391330992, 70598570456104, 1151382852200680, 19835976878704628, 359963038816096924, 6863033015330999110, 137156667020252478684, 2867083618970831936826

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

For n > 1, also the number of balanced reduced multisystems whose atoms are an integer partition of n with at least one part > 1. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. - Gus Wiseman, Dec 31 2019

			The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
  5  (14)  (23)  (113)      (122)      (1112)
                 ((1)(13))  ((1)(22))  ((1)(112))
                 ((3)(11))  ((2)(12))  ((2)(111))
                                       ((11)(12))
                                       ((1)(1)(12))
                                       ((1)(2)(11))
                                       (((1))((1)(12)))
                                       (((1))((2)(11)))
                                       (((2))((1)(11)))
                                       (((12))((1)(1)))
                                       (((11))((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

The maximum-depth case is A000111.

Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.

normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]],{f,Select[facs[n],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n, i, i==1), u=vector(n)); for(r=1, #v, u += v*sum(j=r, #v, (-1)^(j-r)*binomial(j-1, r-1)); v=EulerT(v)); u} \\ Andrew Howroyd, Dec 30 2019

a(n > 1) = A330679(n)/2. - Gus Wiseman, Dec 31 2019

Terms a(14) and beyond from Andrew Howroyd, Dec 30 2019

Terminology corrected by Gus Wiseman, Dec 31 2019

A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 127, 168, 208, 267, 343, 431, 536, 676, 836, 1045, 1283, 1582, 1949, 2395, 2895, 3549, 4298, 5216, 6281, 7569, 9104, 10953, 13078, 15652, 18627, 22207, 26325, 31278, 37002, 43708, 51597, 60807, 71533, 84031

Offset: 0

Gus Wiseman, Mar 15 2025

nonn

			The partition y = (3,3,3,2,2,1,1,1,1) has only one multiset partition into a set of sets, namely {{1},{3},{1,2},{1,3},{1,2,3}}, but this does not have distinct sums, so y is counted under a(17).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
These partitions are ranked by A381806, zeros of A381634 and A381633.
The complement is counted by A381992, ranked by A382075.
For distinct blocks we have A382078, complement A382077, unique A382079.
MM-numbers of these multiset partitions (strict blocks with distinct sum) are A382201.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
Cf. A293243, A293511, A358914, A381078, A381441, A381454.

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[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,10}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A213242 Number of generalizations of the partition 1^n.

Original entry on oeis.org

1, 2, 3, 6, 12, 31, 89, 303, 1119, 4649, 20572, 99241, 502622, 2725840, 15424019, 92211327, 571446565, 3716191974, 24920512847, 174169990243, 1251875604302, 9326245177768, 71241318920624, 562221733320241, 4535497053407716, 37677863148632647, 319551379756283637

Offset: 1

Alois P. Heinz, Jun 14 2012

nonn

Consider the ranked poset L(n) of partitions defined in A002846. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at any node in the poset and end at 1^n.

			For n=5 there are a(5) = 12 paths to 1^5 = 11111: 11111; 2111->11111; 221->2111->11111; 311->2111->11111; 32->221->2111->11111; 32->311->2111->11111; 41->221->2111->11111; 41->311->2111->11111; 5->32->221->2111->11111; 5->32->311->2111->11111; 5->41->221->2111->11111; 5->41->311->2111->11111.

Alois P. Heinz, Table of n, a(n) for n = 1..50
Olivier Gérard, The ranked posets L(2),...,L(8)

Cf. A002846, A213385, A213427, A213597.

b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
      `if`(n<2, 1, `if`(l[n]=0, b(subsop(n=NULL, l)),
      add(`if`(l[i]=0, 0, add(b([seq(l[t]-`if`(t=1, l[t],
      `if`(t=i, 1, `if`(t=j and t=i-j, -2, `if`(t=j or t=i-j,
      -1, 0)))), t=1..n)]), j=1..i/2)), i=2..n)))
    end:
g:= proc(n, i, l)
      `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
      `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=1..25);

b[l_] := b[l] = With[{n = Length[l]}, If[n < 2, 1, If[l[[n]] == 0, b[ReplacePart[l, n -> Sequence[] ]], Sum[If[l[[i]] == 0, 0, Sum[b[Join[Table[l[[t]]-If[t == 1, l[[t]], If[t == i, 1, If[t == j && t == i-j, -2, If[t == j || t == i-j, -1, 0]]]], {t, 1, n}]]], {j, 1, i/2}]], {i, 2, n}]]] ]; g[n_, i_, l_] := If[n == 0 && i == 0, b[l], If[i == 1, b[Prepend[l, n]], Sum[g[n-i*j, i-1, If[l == {} && j == 0, l, Prepend[ l, j]]], {j, 0, n/i}]]] ; a[n_] := g[n, n, {}]; Table[a[n], {n, 1, 27}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-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 06 2025

nonn

First differs from A050326 at a(30) = 4, A050326(30) = 5.
First differs from A339742 at a(42) = 5, A339742(42) = 4.
First differs from A381441 at a(30) = 4, A381441(30) = 5.
First differs from A381633 at a(210) = 10, A381633(210) = 12.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).
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 con 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 with distinct block-sums 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 arrow {1,1,2} -> {4}.

			The prime indices of 120 are {1,1,2,3}, with 3 ways:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
with block-sums: {1,6}, {3,4}, {1,2,4}, so a(120) = 3.
The prime indices of 210 are {1,2,3,4}, with 12 ways:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{1},{2},{3},{4}}
with block-sums: {10}, {1,9}, {2,8}, {3,7}, {4,6}, {3,7}, {4,6}, {1,2,7}, {1,3,6}, {1,4,5}, {2,3,5}, {1,2,3,4}, of which 10 are distinct, so a(210) = 10.

Without distinct block-sums we have A381078 (lower A381454), before sums A050320.
For distinct blocks instead of sums we have A381441, before sums A050326, see A358914.
Before taking sums we had A381633.
Positions of 0 are A381806.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A001055 counts multiset partitions, 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, A116540, A213242, A213385, A213427, A299202, A300385, A317142, A317143, A318360.

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[Union[Sort[hwt/@#]&/@Select[sfacs[n],UnsameQ@@hwt/@#&]]],{n,100}]

A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

First differs from A050361 at a(1728) = 7, A050361(1728) = 8.

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.

			The prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into distinct constant blocks:
  {{2,2,2},{1,1,1,1,1,1}}
  {{1},{2,2,2},{1,1,1,1,1}}
  {{2},{2,2},{1,1,1,1,1,1}}
  {{1,1},{2,2,2},{1,1,1,1}}
  {{1},{2},{2,2},{1,1,1,1,1}}
  {{1},{1,1},{1,1,1},{2,2,2}}
  {{2},{1,1},{2,2},{1,1,1,1}}
  {{1},{2},{1,1},{2,2},{1,1,1}}
with sums:
  {6,6}
  {1,5,6}
  {2,4,6}
  {2,4,6}
  {1,2,4,5}
  {1,2,3,6}
  {2,2,4,4}
  {1,2,2,3,4}
of which 7 are distinct, so a(1728) = 7.

Without distinct blocks (A000688) we have A381455, lower (A355731) A381453.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
Positions of terms > 1 are A046099.
Before taking sums we had A050361.
For equal instead of distinct blocks we have A362421.
For strict instead of constant blocks we have A381441, before sums A050326.
For just distinct blocks we have A381452, before sums A045778.
For distinct sums we have A381716, before sums A381635, zeros A381636.
A001055 counts multiset partitions, 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.
Cf. A000720, A001222, A002846, A005117, A050342, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&&And@@SameQ@@@#&]]],{n,100}]

A213385 a(n) = number of refinements of the partition n^1.

Original entry on oeis.org

1, 2, 3, 7, 15, 43, 131, 468, 1776, 7559, 34022, 166749, 853823, 4682358, 26720781, 161074458, 1004485751, 6576974188, 44322716809, 311440019349, 2247888977510, 16819336465164, 128915407382036, 1021269823516449, 8261243728564640, 68848043979970646

Offset: 1

N. J. A. Sloane, Jun 10 2012

nonn

Consider the ranked poset L(n) of partitions defined in A002846. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at n^1 and end at a node in the poset.

			Referring to the ranked poset L(5) shown in the example in A002846, there are 15 paths that start at ooooo:
end point / number of paths
ooooo / 1
o oooo / 1
oo ooo / 1
o o ooo / 2
o oo oo / 2
o o o oo / 4
o o o o o / 4
Total a(5) = 15.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..50
Olivier Gérard, The ranked posets L(2),...,L(8)
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence labeled H.

Cf. A002846, A213242, A213427.

b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
      `if`(l[n]=1 and {l[1..n-1][]} minus {0}={}, 1,
      add(`if`(l[i]=0, 0, add(`if`(l[j]=0 or i=j and l[j]<2, 0,
      b([seq(`if`(t>n, 0, l[t])-`if`(t=i and t=j, 2, `if`(t=i or t=j,
      1, `if`(t=i+j, -1, 0))), t=1..max(n, i+j))])), j=i..n)), i=1..n))
    end:
g:= proc(n, i, l)
      `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
      `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 11 2012

b[l_List] := b[l] = Module[{n, i, j, t}, n = Length[l]; If[l[[n]] == 1 && Union[ l[[1 ;; n-1]]] ~Complement~ {0} == {}, 1, Sum[If[l[[i]] == 0, 0,  Sum[If[l[[j]] == 0 || i == j && l[[j]]<2, 0, b[Table[If[t>n, 0, l[[t]]] - Which[t == i && t == j, 2, t == i || t == j, 1, t == i+j, -1, True, 0], {t, 1, Max[n, i+j]}]]], {j, i, n}] ], {i, 1, n}]]]; g[n_, i_, l_List] := If[n == 0 && i == 0, b[l], If[i == 1, b[ Join[{n}, l]], Sum[g[n-i*j, i-1, If[l == {} && j == 0, l, Join[{j}, l]]], {j, 0, n/i}]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

Definition clarified by David Applegate, Jun 10 2012
More terms from Alois P. Heinz, Jun 11 2012
Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 14, 56, 219, 1001, 4588

Offset: 0

Gus Wiseman, Jan 27 2019

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Cf. A002846, A005121, A007716, A050343, A213427, A269134, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.

Cf. A323788, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

cp's OEIS Frontend

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

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A300385 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.

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A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

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A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

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A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

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A213242 Number of generalizations of the partition 1^n.

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A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.

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A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

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