A318284 -id:A318284 - cp's OEIS frontend

A318287 Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 2, 3, 2, 3, 4, 5, 3, 7, 4, 7, 9, 5, 5, 12, 6, 12, 14, 10, 8, 13, 12, 14, 14, 18, 10, 34

Offset: 1

Gus Wiseman, Aug 23 2018

			Non-isomorphic representatives of the a(20) = 12 strict multiset partitions of {1,1,1,2,3}:
  {{1,1,1,2,3}}
  {{1},{1,1,2,3}}
  {{2},{1,1,1,3}}
  {{1,1},{1,2,3}}
  {{1,2},{1,1,3}}
  {{2,3},{1,1,1}}
  {{1},{2},{1,1,3}}
  {{1},{1,1},{2,3}}
  {{1},{1,2},{1,3}}
  {{2},{3},{1,1,1}}
  {{2},{1,1},{1,3}}
  {{1},{2},{3},{1,1}}

Cf. A001055, A007716, A045778, A181821, A305936, A316980, A317533, A317776, A317791.

Cf. A318283, A318284, A318285, A318286, A318357, A318362, A318371.

a(n) = A318357(A181821(n)).

A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

Original entry on oeis.org

1, 2, 3, 4, 6, 15, 44

Offset: 1

Gus Wiseman, Jan 07 2020

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 inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
  ()  (2)  (4)    (6)    (12)     (72)
           (2*2)  (2*3)  (2*6)    (8*9)
                         (3*4)    (2*36)
                         (2*2*3)  (3*24)
                                  (4*18)
                                  (6*12)
                                  (2*4*9)
                                  (2*6*6)
                                  (3*3*8)
                                  (3*4*6)
                                  (2*2*18)
                                  (2*3*12)
                                  (2*2*2*9)
                                  (2*2*3*6)
                                  (2*3*3*4)
                                  (2*2*2*3*3)

The same for prime numbers (instead of powers of 2) is A330993,
Factorizations are A001055, with image A045782.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly 2^n factorizations is A330989.
Cf. A033833, A045778, A045783, A181821, A305936, A318283, A318284, A330972, A330973, A330976, A330998, A331022.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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],IntegerQ[Log[2,Length[facs[Times@@Prime/@nrmptn[#]]]]]&]

A001055(A181821(a(n))) = 2^k for some k >= 0.

A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 6, 4, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 84, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 17 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}.

			The a(9) = 6 matrices:
  [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
The a(38) = 9 matrices:
  [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1]
  [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1]
  [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]

The positions of 0's are numbers whose sum of prime indices is not a perfect square (A323527).
The positions of 1's are primes indexed by squares (A323526).
Cf. A008480, A026478, A056239, A103198, A112798, A120732, A318284, A318762, A323519, A323528, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
Table[If[IntegerQ[Sqrt[Total[primeMS[n]]]],Length[Permutations[nrmptn[n]]],0],{n,100}]

If A056239(n) is a perfect square, a(n) = A318762(n). Otherwise, a(n) = 0.

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 8, 7, 7, 8, 11, 12, 15, 5, 15, 17, 22, 14, 27, 19, 30, 13, 27, 30, 33, 26, 42, 37, 56, 7, 44, 45, 51, 34, 77, 67, 72, 25

Offset: 1

Gus Wiseman, Aug 28 2018

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

The prime indices of n are the n-th row of A296150.

			The a(18) = 17 combinatory separations of {1,1,2,2,3}:
  {11223}
  {1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
  {1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
  {1,1,1,11} {1,1,1,12}
  {1,1,1,1,1}

Cf. A007716, A056239, A255906, A269134, A296150, A305936.

Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize,mps[nrmptn[n]],{2}]]],{n,20}]

A320658 Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 5, 2, 1, 3, 0, 0, 0, 1, 0, 6, 1, 0, 2, 4, 0, 0, 1, 0, 0, 1, 0, 9, 3, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0, 1, 1, 6, 15, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 6, 2, 0, 0, 1, 0, 17, 1, 0, 7, 2, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

This multiset 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(84) = 7 factorizations into semiprimes:
  84 = (4*4*9*35)
  84 = (4*4*15*21)
  84 = (4*6*6*35)
  84 = (4*6*10*21)
  84 = (4*6*14*15)
  84 = (4*9*10*14)
  84 = (6*6*10*14)
The a(84) = 7 multiset partitions into pairs:
  {{1,1},{1,1},{2,2},{3,4}}
  {{1,1},{1,1},{2,3},{2,4}}
  {{1,1},{1,2},{1,2},{3,4}}
  {{1,1},{1,2},{1,3},{2,4}}
  {{1,1},{1,2},{1,4},{2,3}}
  {{1,1},{2,2},{1,3},{1,4}}
  {{1,2},{1,2},{1,3},{1,4}}

Cf. A001221, A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320659.

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

A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

This multiset 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(48) = 6 factorizations:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(48) = 6 multiset partitions:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}

A001147(n) = a(2^(2n)).

Cf. A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320658.

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

A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 22 2018

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

Also the coefficient of h(v) in f(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -2  1
  (11):  1
.
  (3):    3 -3  1
  (21):  -3  1
  (111):  1
.
  (4):    -4  2  4 -4  1
  (22):    2  1 -2
  (31):    4 -2 -1  1
  (211):  -4     1
  (1111):  1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):    -5  1  5 -3 -1  1
  (32):    -5  5 -1  1 -2
  (221):    5 -3  1
  (311):    5 -1 -2     1
  (2111):  -5  1
  (11111):  1
For example, row 14 gives: m(32) = -5e(5) - e(32) + 5e(41) + e(221) - 2e(311).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321746.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321912-A321935.

A321918 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 22 2018

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

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -2  1
  (11):     1
.
  (3):    3 -3  1
  (21):     -2  1
  (111):        1
.
  (4):    -4  2  4 -4  1
  (22):       4    -4  1
  (31):          3 -3  1
  (211):           -2  1
  (1111):              1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):       -4     2  4 -4  1
  (32):          -6  6  3 -5  1
  (221):             4    -4  1
  (311):                3 -3  1
  (2111):                 -2  1
  (11111):                    1
For example, row 14 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321752.

Cf. A005651, A008480, A056239, A124794, A124795, A135278, A215366, A318284, A319191, A319193, A319225, A319226, A321912-A321935.

A322454 Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 4, 0, 4, 0, 3, 3, 1, 0, 7, 4, 1, 9, 4, 0, 7, 0, 11, 3, 1, 5, 15, 0, 1, 4, 11

Offset: 1

Gus Wiseman, Dec 09 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(30) = 7 multiset partitions:
    {{1,1,1,2,2,3}}
   {{1,2},{1,1,2,3}}
   {{1,3},{1,1,2,2}}
   {{2,3},{1,1,1,2}}
   {{1,1,2},{1,2,3}}
   {{1,1,3},{1,2,2}}
  {{1,2},{1,2},{1,3}}

Cf. A000688, A000961, A001055, A001597, A023893, A023894, A181821, A318284, A320322, A321407, A321760, A322260, A322452.

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[Length[Select[mps[nrmptn[n]],Min@@Length/@Union/@#>1&]],{n,20}]

A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 3, 1, 7, 7, 1, 5, 5, 1, 5, 9, 5, 1, 9, 11, 1, 9, 28, 36, 16, 1, 10, 24, 16, 1, 14, 38, 27, 1, 13, 18, 1, 13, 69, 160, 164, 61, 1, 24, 79, 62, 1, 20, 160, 580, 1022, 855, 272, 1, 19, 59, 45, 1, 27, 138, 232, 123, 1, 17, 77, 121, 61

Offset: 2

Gus Wiseman, Jan 04 2020

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.

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 whose multiplicities are the prime indices of n (such as 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}.

			Triangle begins:
   {}
   1
   1
   1   1
   1   2
   1   3   2
   1   3
   1   7   7
   1   5   5
   1   5   9   5
   1   9  11
   1   9  28  36  16
   1  10  24  16
   1  14  38  27
   1  13  18
   1  13  69 160 164  61
   1  24  79  62
For example, row n = 12 counts the following multisystems:
  {1,1,2,3}  {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,3}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{3},{1,1,2}}    {{{1}},{{3},{1,2}}}
             {{1},{1},{2,3}}  {{{1,3}},{{1},{2}}}
             {{1},{2},{1,3}}  {{{2}},{{1},{1,3}}}
             {{1},{3},{1,2}}  {{{2}},{{3},{1,1}}}
             {{2},{3},{1,1}}  {{{2,3}},{{1},{1}}}
                              {{{3}},{{1},{1,2}}}
                              {{{3}},{{2},{1,1}}}

Row sums are A318846.
Final terms in each row are A330728.
Row prime(n) is row n of A330784.
Row 2^n is row n of A008826.
Row n is row A181821(n) of A330667.
Column k = 3 is A318284(n) - 2 for n > 2.
Cf. A000111, A002846, A005121, A292504, A318812, A318813, A318847, A318848, A318849, A330475, A330666, A330935.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[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[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

T(2^n,k) = A008826(n,k).

cp's OEIS Frontend

A318287 Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

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A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

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A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

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A320658 Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs.

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A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.

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A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

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A321918 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

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A322454 Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.

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