A318284 -id:A318284 - cp's OEIS frontend

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Gus Wiseman, Jan 07 2020

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

The same for powers of 2 (instead of primes) is A330990.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Factorizations of a number's inverse prime shadow are A318284.
Numbers with a prime number of factorizations are A330991.
Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

A001055(A181821(a(n))) belongs to A000040.

A339842 Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.

Original entry on oeis.org

9, 25, 30, 49, 63, 70, 75, 84, 100, 121, 147, 154, 165, 169, 175, 189, 196, 198, 210, 220, 250, 264, 273, 280, 286, 289, 325, 343, 351, 361, 363, 364, 385, 390, 441, 442, 462, 468, 484, 490, 495, 507, 520, 525, 529, 550, 561, 588, 594, 595, 616, 624, 637, 646

Offset: 1

Gus Wiseman, Dec 27 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph, and multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      9: {2,2}        189: {2,2,2,4}      363: {2,5,5}
     25: {3,3}        196: {1,1,4,4}      364: {1,1,4,6}
     30: {1,2,3}      198: {1,2,2,5}      385: {3,4,5}
     49: {4,4}        210: {1,2,3,4}      390: {1,2,3,6}
     63: {2,2,4}      220: {1,1,3,5}      441: {2,2,4,4}
     70: {1,3,4}      250: {1,3,3,3}      442: {1,6,7}
     75: {2,3,3}      264: {1,1,1,2,5}    462: {1,2,4,5}
     84: {1,1,2,4}    273: {2,4,6}        468: {1,1,2,2,6}
    100: {1,1,3,3}    280: {1,1,1,3,4}    484: {1,1,5,5}
    121: {5,5}        286: {1,5,6}        490: {1,3,4,4}
    147: {2,4,4}      289: {7,7}          495: {2,2,3,5}
    154: {1,4,5}      325: {3,3,6}        507: {2,6,6}
    165: {2,3,5}      343: {4,4,4}        520: {1,1,1,3,6}
    169: {6,6}        351: {2,2,2,6}      525: {2,3,3,4}
    175: {3,3,4}      361: {8,8}          529: {9,9}
For example, a complete list of all multigraphs with degrees (4,2,2,2) is:
  {{1,2},{1,2},{1,3},{1,4},{3,4}}
  {{1,2},{1,3},{1,3},{1,4},{2,4}}
  {{1,2},{1,3},{1,4},{1,4},{2,3}}
Since none of these is strict, i.e., a graph, the Heinz number 189 is in the sequence.

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross references.
Distinct prime shadows (images under A181819) of A340017.
A000070 counts non-multigraphical partitions (A339620).
A000569 counts graphical partitions (A320922).
A027187 counts partitions of even length (A028260).
A058696 counts partitions of even numbers (A300061).
A096373 cannot be partitioned into strict pairs.
A209816 counts multigraphical partitions (A320924).
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A320893 can be partitioned into distinct pairs but not into strict pairs.
A339560 can be partitioned into distinct strict pairs.
A339617 counts non-graphical partitions of 2n (A339618).
A339659 counts graphical partitions of 2n into k parts.
Cf. A004251, A006129, A007717, A056239, A095268, A112798, A181821, A305936, A318284, A339559.

strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&& Select[strr[Times@@Prime/@nrmptn[#]],UnsameQ@@#&]=={}&&strr[Times@@Prime/@nrmptn[#]]!={}&]

Equals A320924 /\ A339618.

Equals A320924 \ A320922.

A321931 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 2, -3, 1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 2, -1, -2, 1, 0, -6, 3, 8, -6, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 2, -1, -2, 1, 0, 0, 0, 2, -2, -1, 0, 1, 0, 0, -6, 6, 5, -3, -3, 1, 0

Offset: 1

Gus Wiseman, Nov 23 2018

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

The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1
  (11): -1  1
.
  (3):    1
  (21):  -1  1
  (111):  2 -3  1
.
  (4):     1
  (22):   -1  1
  (31):   -1     1
  (211):   2 -1 -2  1
  (1111): -6  3  8 -6  1
.
  (5):      1
  (41):    -1  1
  (32):    -1     1
  (221):    2 -1 -2  1
  (311):    2 -2 -1     1
  (2111):  -6  6  5 -3 -3  1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: M(32) = -p(5) + p(32).

Wikipedia, Symmetric polynomial

Row sums are A155972. This is a regrouping of the triangle A321895.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 36, 0, 14, 0, 0, 0, 25, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 0, 0, 0, 75

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} with x < z < y < t or z < x < t < y.

			The a(36) = 10 crossing multiset partitions of {1,1,2,2,3,4}:
  {{1,3},{1,2,2,4}}
  {{2,4},{1,1,2,3}}
  {{1,1,3},{2,2,4}}
  {{1,2,3},{1,2,4}}
  {{1},{1,3},{2,2,4}}
  {{1},{2,4},{1,2,3}}
  {{2},{1,3},{1,2,4}}
  {{2},{1,1,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1},{2},{1,3},{2,4}}

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324325.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324325(n) = A318284(n).

A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 5, 7, 9, 11, 7, 11, 11, 15, 7, 12, 16, 21, 16, 26, 26, 36, 12, 21, 26, 36, 21, 36, 36, 52, 11, 19, 29, 38, 31, 52, 52, 74, 29, 52, 66, 92, 52, 92, 92, 135, 19, 38, 52, 74, 52, 92, 92, 135, 38, 74, 92, 135, 74, 135, 135, 203, 15, 30, 47

Offset: 0

Gus Wiseman, Apr 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The a(1) = 1 through a(11) = 11 multiset partitions:
  {1}  {11}    {12}    {111}      {112}      {122}      {123}
       {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{22}    {1}{23}
                       {1}{1}{1}  {2}{11}    {2}{12}    {2}{13}
                                  {1}{1}{2}  {1}{2}{2}  {3}{12}
                                                        {1}{2}{3}
  {1111}        {1112}        {1122}        {1123}
  {1}{111}      {1}{112}      {1}{122}      {1}{123}
  {11}{11}      {11}{12}      {11}{22}      {11}{23}
  {1}{1}{11}    {2}{111}      {12}{12}      {12}{13}
  {1}{1}{1}{1}  {1}{1}{12}    {2}{112}      {2}{113}
                {1}{2}{11}    {1}{1}{22}    {3}{112}
                {1}{1}{1}{2}  {1}{2}{12}    {1}{1}{23}
                              {2}{2}{11}    {1}{2}{13}
                              {1}{1}{2}{2}  {1}{3}{12}
                                            {2}{3}{11}
                                            {1}{1}{2}{3}

The described multiset has A000120 distinct parts.
The sum of the described multiset is A029931.
Multisets of compositions are A034691.
The described multiset is a row of A095684.
Combinatory separations of normal multisets are A269134.
The product of the described multiset is A284001.
The version for prime indices is A318284.
The version counting combinatory separations is A334030.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Length of Lyndon factorization is A329312.
- Dealings are counted by A333939.
- Distinct parts are counted by A334028.
- Length of co-Lyndon factorization is A334029.
Cf. A057335, A065609, A275692, A292884, A318560, A318563, A326774, A333764, A333765, A333940.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[Times@@Prime/@ptnToNorm[stc[n]]]],{n,0,30}]

a(n) = A001055(A057335(n)).

A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

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

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(180) = 4 strict set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2},{3},{1,2}}

Cf. A007716, A045778, A049311, A050326, A056239, A112798, A116539, A116540, A292432, A292444, A293511, A318284, A318360, A318369.

A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, -1, 0, 2, 1, 1, 1, -2, 0, 1, 0, 0, 0, 2, 0, -2, -2, -1, 1, -1, -2, 3, -1, 1, -2, -3, -2, 3, 0, -3, 1, -4, -5, 1, -1, -2, -1, 5, -5, 1, -3, 1, -1, -5, -4, 5, 1, -1, -9, -2, -1, -6, -1, -3, -2, 7, -7, -8, -2, -2

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320836.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
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[Sum[(-1)^(Length[m]-1),{m,mps[nrmptn[n]]}],{n,30}]

a(n) = A316441(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -2, -1, 0, -2, 0, -2, -1, -1, -1, -4, -1, -1, -1, -3, 0, -3, 0, -2, -4, -1, -1, -6, -2, -3, -2, -2, 0, -6, -2, -4, -1, -1, 0, -5, 0, -1, -3, -9, -2, -3, 0, -2, -1, -3, 0, -7, 0

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320835.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
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[Sum[(-1)^Length[m],{m,Select[mps[nrmptn[n]],UnsameQ@@#&]}],{n,30}]

a(n) = A114592(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A321270 Number of connected multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 5, 4, 7, 3, 11, 7, 10, 1, 15, 9, 22, 7, 19, 12, 30, 5, 22, 19, 28, 14, 42, 22, 56, 1, 33, 30, 42, 20, 77, 45

Offset: 1

Gus Wiseman, Nov 01 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(2) = 1 through a(12) = 3 connected multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}          {{1123}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}        {{1}{123}}
           {{12}{12}}    {{11}{12}}    {{11}{111}}        {{12}{13}}
           {{2}{112}}    {{1}{1}{12}}  {{1}{1}{111}}
           {{1}{2}{12}}                {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
The a(18) = 9, a(27) = 28, and a(36) = 20 connected multiset partitions of {1,1,2,2,3}, {1,1,2,2,3,3}, and {1,1,2,2,3,4} respectively:
  {{1,1,2,2,3}}      {{1,1,2,2,3,3}}        {{1,1,2,2,3,4}}
  {{1},{1,2,2,3}}    {{1},{1,2,2,3,3}}      {{1},{1,2,2,3,4}}
  {{1,2},{1,2,3}}    {{1,1,2},{2,3,3}}      {{1,1,2},{2,3,4}}
  {{1,3},{1,2,2}}    {{1,1,3},{2,2,3}}      {{1,2},{1,2,3,4}}
  {{2},{1,1,2,3}}    {{1,2},{1,2,3,3}}      {{1,2,2},{1,3,4}}
  {{2,3},{1,1,2}}    {{1,2,2},{1,3,3}}      {{1,2,3},{1,2,4}}
  {{1},{1,2},{2,3}}  {{1,2,3},{1,2,3}}      {{1,3},{1,2,2,4}}
  {{1},{2},{1,2,3}}  {{1,3},{1,2,2,3}}      {{1,4},{1,2,2,3}}
  {{2},{1,2},{1,3}}  {{2},{1,1,2,3,3}}      {{2},{1,1,2,3,4}}
                     {{2,3},{1,1,2,3}}      {{2,3},{1,1,2,4}}
                     {{3},{1,1,2,2,3}}      {{2,4},{1,1,2,3}}
                     {{1},{1,2},{2,3,3}}    {{1},{1,2},{2,3,4}}
                     {{1},{1,3},{2,2,3}}    {{1},{2},{1,2,3,4}}
                     {{1},{2},{1,2,3,3}}    {{1,2},{1,3},{2,4}}
                     {{1,2},{1,3},{2,3}}    {{1,2},{1,4},{2,3}}
                     {{1},{2,3},{1,2,3}}    {{1},{2,3},{1,2,4}}
                     {{1},{3},{1,2,2,3}}    {{1},{2,4},{1,2,3}}
                     {{2},{1,2},{1,3,3}}    {{2},{1,2},{1,3,4}}
                     {{2},{1,3},{1,2,3}}    {{2},{1,3},{1,2,4}}
                     {{2},{2,3},{1,1,3}}    {{2},{1,4},{1,2,3}}
                     {{2},{3},{1,1,2,3}}
                     {{3},{1,2},{1,2,3}}
                     {{3},{1,3},{1,2,2}}
                     {{3},{2,3},{1,1,2}}
                     {{1},{2},{1,3},{2,3}}
                     {{1},{2},{3},{1,2,3}}
                     {{1},{3},{1,2},{2,3}}
                     {{2},{3},{1,2},{1,3}}

Cf. A007718, A007719, A056156, A181821, A191970, A300913, A305193, A305936, A318284, A318286, A319557, A321272.

A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 1, 4, 4, 7, 3, 11, 7, 8, 1, 15, 8, 22, 7, 14, 12, 30, 5, 16, 19, 20, 14, 42, 18, 56, 1, 24, 30, 28, 18, 77, 45, 38, 14

Offset: 1

Gus Wiseman, Nov 01 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(15) = 8 multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}
           {{2}{112}}    {{11}{12}}    {{11}{111}}
           {{1}{2}{12}}  {{1}{1}{12}}  {{1}{1}{111}}
                                       {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
.
  {{1123}}    {{111111}}            {{11112}}        {{11122}}
  {{1}{123}}  {{1}{11111}}          {{1}{1112}}      {{1}{1122}}
  {{12}{13}}  {{11}{1111}}          {{11}{112}}      {{11}{122}}
              {{111}{111}}          {{12}{111}}      {{2}{1112}}
              {{1}{1}{1111}}        {{1}{1}{112}}    {{1}{1}{122}}
              {{1}{11}{111}}        {{1}{11}{12}}    {{1}{2}{112}}
              {{11}{11}{11}}        {{1}{1}{1}{12}}  {{2}{11}{12}}
              {{1}{1}{1}{111}}                       {{1}{1}{2}{12}}
              {{1}{1}{11}{11}}
              {{1}{1}{1}{1}{11}}
              {{1}{1}{1}{1}{1}{1}}

Cf. A007718, A181821, A303837, A304382, A305081, A305936, A318284, A321155, A321229, A321253, A321270, A321271.

a(prime(n)) = A000041(n).

cp's OEIS Frontend

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

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A339842 Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.

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A321931 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.

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A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

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A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

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A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

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A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

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