A181821 -id:A181821 - cp's OEIS frontend

A318284 Number of multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 15, 15, 26, 22, 21, 29, 19, 30, 36, 31, 30, 66, 38, 42, 52, 56, 52, 47, 45, 57, 92, 77, 67, 77, 74, 101, 98, 135, 64, 137, 97, 176, 135, 109, 109, 118, 105, 231, 249, 97, 141, 181, 139, 297, 198, 385, 195, 269

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

			The a(12) = 11 multiset partitions of {1,1,2,3}:
  {{1,1,2,3}}
  {{1},{1,2,3}}
  {{2},{1,1,3}}
  {{3},{1,1,2}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{2},{3},{1,1}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..700

Cf. A001055, A001970, A007716, A181821, A219727, A255906, A269134, A305936.

Cf. A317532, A318283, A318285, A318286, A318360, A318361.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{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]]}]];
Table[Length[facs[Times@@Prime/@nrmptn[n]]],{n,60}]

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), A=O(x*x^vecmax(sig)), s=0); forpart(p=n, my(q=1/prod(i=1, #p, 1 - x^p[i] + A)); 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==1, numbpart(s[1]), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018

a(n) = A001055(A181821(n)).

a(prime(n)^k) = A219727(n,k). - Andrew Howroyd, Dec 10 2018

A305936 Irregular triangle whose n-th row is the multiset spanning an initial interval of positive integers with multiplicities equal to the n-th row of A296150 (the prime indices of n in weakly decreasing order).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 23 2018

			Row 90 is {1,1,1,2,2,3,3,4} because 90 = prime(3)*prime(2)*prime(2)*prime(1).
Triangle begins:
   1:
   2:  1
   3:  1  1
   4:  1  2
   5:  1  1  1
   6:  1  1  2
   7:  1  1  1  1
   8:  1  2  3
   9:  1  1  2  2
  10:  1  1  1  2
  11:  1  1  1  1  1
  12:  1  1  2  3
  13:  1  1  1  1  1  1

Row lengths are A056239. Number of distinct elements in row n is A001222(n). Number of distinct multiplicities in row n is A001221(n).
Cf. A000041, A000720, A112798, A181821, A182850, A255906, A296150, A304464.
Cf. A318283, A318284, A318285, A318286, A318287, A318360, A318361, A318362, A318371.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Array[nrmptn,30]

A325277 Irregular triangle read by rows where row 1 is {1} and row n is the sequence starting with n and repeatedly applying A181819 until a prime number is reached.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 4, 3, 7, 8, 5, 9, 3, 10, 4, 3, 11, 12, 6, 4, 3, 13, 14, 4, 3, 15, 4, 3, 16, 7, 17, 18, 6, 4, 3, 19, 20, 6, 4, 3, 21, 4, 3, 22, 4, 3, 23, 24, 10, 4, 3, 25, 3, 26, 4, 3, 27, 5, 28, 6, 4, 3, 29, 30, 8, 5, 31, 32, 11, 33, 4, 3

Offset: 1

Gus Wiseman, Apr 15 2019

The function A181819 maps p^i*...*q^j to prime(i)*...*prime(j) where p through q are distinct primes.

			Triangle begins:
   1            26 4 3        51 4 3          76 6 4 3
   2            27 5          52 6 4 3        77 4 3
   3            28 6 4 3      53              78 8 5
   4 3          29            54 10 4 3       79
   5            30 8 5        55 4 3          80 14 4 3
   6 4 3        31            56 10 4 3       81 7
   7            32 11         57 4 3          82 4 3
   8 5          33 4 3        58 4 3          83
   9 3          34 4 3        59              84 12 6 4 3
  10 4 3        35 4 3        60 12 6 4 3     85 4 3
  11            36 9 3        61              86 4 3
  12 6 4 3      37            62 4 3          87 4 3
  13            38 4 3        63 6 4 3        88 10 4 3
  14 4 3        39 4 3        64 13           89
  15 4 3        40 10 4 3     65 4 3          90 12 6 4 3
  16 7          41            66 8 5          91 4 3
  17            42 8 5        67              92 6 4 3
  18 6 4 3      43            68 6 4 3        93 4 3
  19            44 6 4 3      69 4 3          94 4 3
  20 6 4 3      45 6 4 3      70 8 5          95 4 3
  21 4 3        46 4 3        71              96 22 4 3
  22 4 3        47            72 15 4 3       97
  23            48 14 4 3     73              98 6 4 3
  24 10 4 3     49 3          74 4 3          99 6 4 3
  25 3          50 6 4 3      75 6 4 3       100 9 3

Row lengths are 1 for n = 1 and A323014(n) for n > 1.

Cf. A001221, A001222, A071625, A118914, A181819, A181821, A182850, A182857, A323022, A323023, A325238, A325239.

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
Table[NestWhileList[red,n,#>1&&!PrimeQ[#]&],{n,30}]

T(n,k) = A325239(n,k) for k <= A323014(n).

A001222(T(n,k)) = A323023(n,k) for n > 1.

A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103, 104, 106

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is 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 sequence of terms together with the corresponding multisets begins:
   3: {1,1}
   5: {1,1,1}
   7: {1,1,1,1}
  10: {1,1,1,2}
  11: {1,1,1,1,1}
  13: {1,1,1,1,1,1}
  14: {1,1,1,1,2}
  17: {1,1,1,1,1,1,1}
  19: {1,1,1,1,1,1,1,1}
  21: {1,1,1,1,2,2}
  22: {1,1,1,1,1,2}
  23: {1,1,1,1,1,1,1,1,1}
  26: {1,1,1,1,1,1,2}
  28: {1,1,1,1,2,3}
  29: {1,1,1,1,1,1,1,1,1,1}

The complement is A335127.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A025487, A056239, A106351, A112798, A114938, A181819, A181821, A278990, A292884, A335407, A335489, A335516, A335838.

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],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]=={}&]

A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 5, 1, 0, 0, 4, 0, 0, 0, 15, 0, 5, 0, 1, 0, 0, 0, 16, 0, 0, 8, 0, 0, 2, 0, 52, 0, 0, 0, 23, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 68, 0, 1, 0, 0, 0, 40, 0, 1, 0, 0, 0, 14, 0, 0, 1, 203, 0, 0, 0, 0, 0, 0, 0, 111, 0, 0, 4, 0, 0, 0, 0, 41, 80, 0, 0

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(24) = 16 sets of sets with multiset union {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1,3},{1,2,4}}
  {{1,4},{1,2,3}}
  {{1},{2},{1,3,4}}
  {{1},{3},{1,2,4}}
  {{1},{4},{1,2,3}}
  {{1},{1,2},{3,4}}
  {{1},{1,3},{2,4}}
  {{1},{1,4},{2,3}}
  {{2},{1,3},{1,4}}
  {{3},{1,2},{1,4}}
  {{4},{1,2},{1,3}}
  {{1},{2},{3},{1,4}}
  {{1},{2},{4},{1,3}}
  {{1},{3},{4},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A007716, A045778, A049311, A050326, A116540, A292432, A292444, A293511.

Cf. A318283, A318284, A318286, A318287, A318360, A318361, A318370, A318371.

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,90}]

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(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, s[1]==1, count(sig(n))))} \\ Andrew Howroyd, Dec 18 2018

a(n) = A050326(A181821(n)).

a(prime(n)^k) = A188445(n, k). - Andrew Howroyd, Dec 17 2018

A325238 First positive integer with each omega-sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 96, 120, 128, 192, 210, 216, 240, 256, 360, 384, 420, 480, 512, 720, 768, 840, 900, 960, 1024, 1260, 1296, 1440, 1536, 1680, 1920, 2048, 2310, 2520, 2880, 3072, 3360, 3840, 4096, 4620, 5040, 5760, 6144, 6720

Offset: 1

Gus Wiseman, Apr 14 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = frequency depth of n, and the k-th part is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, given by red(n = p^i*...*q^j) = prime(i)*...*prime(j), i.e., the product of primes indexed by the prime exponents of n.

			The sequence of terms together with their omega-sequences begins:
    1:
    2: 1
    4: 2 1
    6: 2 2 1
    8: 3 1
   12: 3 2 2 1
   16: 4 1
   24: 4 2 2 1
   30: 3 3 1
   32: 5 1
   36: 4 2 1
   48: 5 2 2 1
   60: 4 3 2 2 1
   64: 6 1
   96: 6 2 2 1
  120: 5 3 2 2 1
  128: 7 1
  192: 7 2 2 1
  210: 4 4 1
  216: 6 2 1
  240: 6 3 2 2 1
  256: 8 1
  360: 6 3 3 1
  384: 8 2 2 1
  420: 5 4 2 2 1

Cf. A001221, A001222, A007916, A011784, A070175, A071625, A118914, A181819, A181821, A303555, A304465, A323014, A323023, A325238, A325239.

tomseq[n_]:=If[n<=1,{},Most[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]];
omseqs=Table[Total/@tomseq[n],{n,1000}];
Sort[Table[Position[omseqs,x][[1,1]],{x,Union[omseqs]}]]

A304660 A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).

Original entry on oeis.org

1, 2, 4, 6, 8, 18, 16, 30, 36, 54, 32, 150, 64, 162, 108, 210, 128, 450, 256, 750, 324, 486, 512, 1470, 216, 1458, 900, 3750, 1024, 2250, 2048, 2310, 972, 4374, 648, 7350, 4096, 13122, 2916, 10290, 8192, 11250, 16384, 18750, 4500, 39366, 32768, 25410, 1296

Offset: 1

Gus Wiseman, May 16 2018

nonn

A permutation of A133808. a(n) is the smallest member m of A133808 such that A181819(m) = n.

			Sequence of normalized prime multisets together with the normalized prime multisets of their images begins:
   1:        {} -> {}
   2:       {1} -> {1}
   3:       {2} -> {1,1}
   4:     {1,1} -> {1,2}
   5:       {3} -> {1,1,1}
   6:     {1,2} -> {1,2,2}
   7:       {4} -> {1,1,1,1}
   8:   {1,1,1} -> {1,2,3}
   9:     {2,2} -> {1,1,2,2}
  10:     {1,3} -> {1,2,2,2}
  11:       {5} -> {1,1,1,1,1}
  12:   {1,1,2} -> {1,2,3,3}
  13:       {6} -> {1,1,1,1,1,1}
  14:     {1,4} -> {1,2,2,2,2}
  15:     {2,3} -> {1,1,2,2,2}
  16: {1,1,1,1} -> {1,2,3,4}
  17:       {7} -> {1,1,1,1,1,1,1}
  18:   {1,2,2} -> {1,2,2,3,3}

Cf. A055932, A056239, A112798, A130091, A133808, A181819, A181821, A182850, A182857, A275870, A304455.

Table[With[{y=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Times@@Power[Array[Prime,Length[y]],y]],{n,100}]

a(n) = Product_{i = 1..Omega(n)} prime(i)^A112798(n,i).

A332642 Numbers whose negated unsorted prime signature is not unimodal.

Original entry on oeis.org

90, 126, 198, 234, 270, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 594, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1188, 1206, 1242, 1274, 1275, 1278, 1314, 1350, 1386

Offset: 1

Gus Wiseman, Feb 28 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The sequence of terms together with their prime indices begins:
    90: {1,2,2,3}
   126: {1,2,2,4}
   198: {1,2,2,5}
   234: {1,2,2,6}
   270: {1,2,2,2,3}
   306: {1,2,2,7}
   342: {1,2,2,8}
   350: {1,3,3,4}
   378: {1,2,2,2,4}
   414: {1,2,2,9}
   522: {1,2,2,10}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
   550: {1,3,3,5}
   558: {1,2,2,11}
   594: {1,2,2,2,5}
   630: {1,2,2,3,4}
   650: {1,3,3,6}
   666: {1,2,2,12}
   702: {1,2,2,2,6}
For example, 630 has negated unsorted prime signature (-1,-2,-1,-1), which is not unimodal, so 630 is in the sequence.

Eric Weisstein's World of Mathematics, Unimodal Sequence

These are the Heinz numbers of the partitions counted by A332639.
The case that is not unimodal either is A332643.
The version for compositions is A332669.
The complement is A332282.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
The number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n is A332742(n).
Partitions whose negated 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Cf. A007052, A056239, A112798, A181821, A242031, A329747, A332280, A332281, A332578, A332671, A332831.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[2000],!unimodQ[-Last/@FactorInteger[#]]&]

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 2, 4, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 1, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 1, 2, 1, 6, 2, 2

Offset: 0

Gus Wiseman, Apr 09 2025

nonn

The first x with a(x) > 1 but A382771(x) > 0 is a(216) = 4, A382771(216) = 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, sum A056239.

			The prime indices of 216 are {1,1,1,2,2,2} and we have permutations:
  (1,1,1,2,2,2)
  (1,2,1,2,1,2)
  (2,1,2,1,2,1)
  (2,2,2,1,1,1)
so a(216) = 4.
The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
  (1,2,1,2,1,2,1,2,1,3,1)
  (1,2,1,2,1,2,1,3,1,2,1)
  (1,2,1,2,1,3,1,2,1,2,1)
  (1,2,1,3,1,2,1,2,1,2,1)
  (1,3,1,2,1,2,1,2,1,2,1)
so a(25920) = 5.

The restriction to signature representatives (A181821) is A382858, distinct A382773.
The restriction to factorials is A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382771.
For run-sums instead of run-lengths we have A382877, distinct A382876.
Positions of first appearances are A382878.
Positions of 0 are A382879.
Positions of terms > 1 are A383089.
Positions of 1 are A383112.
A003963 gives product of prime indices.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary expansion has all equal run-lengths, distinct A328592.
A353744 ranks compositions with equal run-lengths, counted by A329738.
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A238130, A238279, A351201, A351293, A351295.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 7, 2, 0, 0, 1, 0, 1, 12, 1, 0, 0, 0, 1, 0, 1, 17, 2, 1, 0, 0, 0, 1, 0, 1, 24, 4, 0, 0, 0, 0, 0, 1, 0, 1, 33, 5, 1, 1, 0, 0, 0, 0, 1, 0, 1, 44, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 57, 14, 3, 0, 1

Offset: 0

Gus Wiseman, Apr 18 2019

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. The omicron of the partition is 0 if the omega-sequence is empty, 1 if it is a singleton, and otherwise the second-to-last part. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), and its omicron is 2.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  5  0  0  1
  0  1  7  2  0  0  1
  0  1 12  1  0  0  0  1
  0  1 17  2  1  0  0  0  1
  0  1 24  4  0  0  0  0  0  1
  0  1 33  5  1  1  0  0  0  0  1
  0  1 44  9  1  0  0  0  0  0  0  1
  0  1 57 14  3  0  1  0  0  0  0  0  1
  0  1 76 20  3  0  0  0  0  0  0  0  0  1
Row n = 8 counts the following partitions.
  (8)  (44)       (431)  (2222)  (11111111)
       (53)       (521)
       (62)
       (71)
       (332)
       (422)
       (611)
       (3221)
       (3311)
       (4211)
       (5111)
       (22211)
       (32111)
       (41111)
       (221111)
       (311111)
       (2111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041. Column k = 2 is A325267.
Cf. A181819, A181821, A304634, A304636, A323014, A323023, A325250, A325273, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==k&]],{n,0,10},{k,0,n}]

omicron(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); r=#p; for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L); r)}
row(n)={my(v=vector(1+n)); forpart(p=n, v[1 + omicron(Vec(p))]++); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

cp's OEIS Frontend

A318284 Number of multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A305936 Irregular triangle whose n-th row is the multiset spanning an initial interval of positive integers with multiplicities equal to the n-th row of A296150 (the prime indices of n in weakly decreasing order).

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A325277 Irregular triangle read by rows where row 1 is {1} and row n is the sequence starting with n and repeatedly applying A181819 until a prime number is reached.

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A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

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A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

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A325238 First positive integer with each omega-sequence.

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A304660 A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).

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A332642 Numbers whose negated unsorted prime signature is not unimodal.

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A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.