A182850 -id:A182850 - cp's OEIS frontend

A318283 Sum of elements of the multiset spanning an initial interval of positive integers with multiplicities equal to the prime indices of n in weakly decreasing order.

0, 1, 2, 3, 3, 4, 4, 6, 6, 5, 5, 7, 6, 6, 7, 10, 7, 9, 8, 8, 8, 7, 9, 11, 9, 8, 12, 9, 10, 10, 11, 15, 9, 9, 10, 13, 12, 10, 10, 12, 13, 11, 14, 10, 13, 11, 15, 16, 12, 12, 11, 11, 16, 16, 11, 13, 12, 12, 17, 14, 18, 13, 14, 21, 12, 12, 19, 12, 13, 13, 20, 18

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

			The multiset spanning an initial interval of positive integers with multiplicities equal to the prime indices of 90 in weakly decreasing order is {1,1,1,2,2,3,3,4}, so a(90) = 1+1+1+2+2+3+3+4 = 17.

Row sums of A305936.

Cf. A000041, A000720, A001221, A001222, A056239, A112798, A181821, A182850, A296150, A304464.

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

a(n) = A056239(A181821(n)).

A182857 Smallest number that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

1, 3, 4, 6, 12, 60, 2520, 1286485200, 35933692027611398678865941374040400000

Offset: 0

Matthew Vandermast, Jan 05 2011

nonn

a(9) has 296 digits.
Related to Levine's sequence (A011784): A011784(n) = A001222(a(n)) = A001221(a(n+1)) = A051903(a(n+2)) = A071625(a(n+2)). Also see A182858.
Values of n where A182850(n) increases to a record.
The multiplicity of prime(k) in a(n+1) is the k-th largest prime index of a(n), which is A296150(a(n),k). - Gus Wiseman, May 13 2018

			From _Gus Wiseman_, May 13 2018: (Start)
Like A001462 the following sequence of multisets whose Heinz numbers belong to this sequence is a run-length describing sequence, as the number of k's in row n + 1 is equal to the k-th term of row n.
{2}
{1,1}
{1,2}
{1,1,2}
{1,1,2,3}
{1,1,1,2,2,3,4}
{1,1,1,1,2,2,2,3,3,4,4,5,6,7}
{1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7,7,7,8,8,9,9,10,10,11,12,13,14}
(End)

Gus Wiseman, Table of n, a(n) for n = 0..9

Cf. A001462, A007755, A007916, A009287, A012257, A112798, A181819, A182850-A182858, A296150, A304455, A304464, A304465.

Prepend[Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{Reverse[#][[i]]}],{i,Length[#]}]&,{2},8],1] (* Gus Wiseman, May 13 2018 *)

For n > 0, a(n) = A181819(a(n+1)). For n > 1, a(n) = A181821(a(n-1)).

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]

A304465 If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 08 2018

			Starting with the multiset of prime factors of 2520 we have {2,2,2,3,3,5,7} -> {1,1,2,3} -> {1,1,2} -> {1,2} -> {1,1} -> {2}, so a(2520) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A001597, A005117, A007916, A055932, A056239, A112798, A181819, A182850, A182857, A275870, A296150, A303945, A304464.

Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}]

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A304465(n) = if(1==n,0,my(t=isprimepower(n)); if(t,t, t=omega(n); if(bigomega(n)==t),t,A304465(A181819(n)))); \\ Antti Karttunen, Nov 08 2018

a(p^n) = n where p is any prime number.
a(product of n distinct primes) = n.
a(1) = 0; and for n > 1, if n = prime^k, a(n) = k, otherwise, if n is squarefree [i.e., A001221(n) = A001222(n)], a(n) = A001221(n), otherwise a(n) = a(A181819(n)). - Antti Karttunen, Nov 08 2018

More terms from Antti Karttunen, Nov 08 2018

A225485 Number of partitions of n that have frequency depth k, an array read by rows.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 4, 3, 1, 1, 4, 8, 1, 1, 3, 6, 9, 3, 1, 2, 8, 12, 7, 1, 3, 11, 17, 10, 1, 1, 11, 26, 17, 1, 5, 19, 25, 27, 1, 1, 17, 44, 38, 1, 3, 25, 53, 52, 1, 1, 3, 29, 63, 76, 4

Offset: 1

Clark Kimberling, May 08 2013

Let S = {x(1),...,x(k)} be a multiset whose distinct elements are y(1),...,y(h).  Let f(i) be the frequency of y(i) in S.  Define F(S) = {f(1),..,f(h)}, F(1,S) = F(S), and F(m,S) = F(F(m-1),S) for m>1.  Then lim(F(m,S)) = {1} for every S, so that there is a least positive integer i for which F(i,S) = {1}, which we call the frequency depth of S.
Equivalently, the frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). - Gus Wiseman, Apr 19 2019
From Clark Kimberling, Sep 26 2023: (Start)
Below, m^n abbreviates the sum m+...+m of n terms. In the following list, the numbers p_1,...,p_k are distinct, m >= 1, and k >= 1. The forms of the partitions being counted are as follows:
   column 1: [n],
   column 2: [m^k],
   column 3: [p_1^m,...,p_k^m],
   column 4: [(p_1^m_1)^m,..., (p_k^m_k)^m], distinct numbers m_i.
Column 3 is of special interest. Assume first that m = 1, so that the form of partition being counted is p = [p_1,...,p_k], with conjugate given by [q_1,...,q_m] where  q_i is the number of parts of p that are >= i. Since the p_i are distinct, the distinct parts of q are the integers 1,2,...,k. For the general case that m >= 1, the distinct parts of q are the integers m,...,km. Let S(n) denote the set of partitions of n counted by column 3. Then if a and b are in the set S*(n) of conjugates of partitions in S(n), and if a > b, then a - b is also in S*(n). Call this the subtraction property. Conversely, if a partition q has the subtraction property, then q must consist of a set of numbers m,..,km for some m. Thus, column 3 counts the partitions of n that have the subtraction property. (End)

			The first 9 rows:
  n = 1 .... 0
  n = 2 .... 1..1
  n = 3 .... 1..1..1
  n = 4 .... 1..2..1..1
  n = 5 .... 1..1..2..3
  n = 6 .... 1..3..4..3
  n = 7 .... 1..1..4..8..1
  n = 8 .... 1..3..6..9..3
  n = 9 .... 1..2..8.12..7
For the 7 partitions of 5, successive frequencies are shown here:
  5 -> 1 (depth 1)
  41 -> 11 -> 2 -> 1 (depth 3)
  32 -> 11 -> 2 -> 1 (depth 3)
  311 -> 12 -> 11 -> 2 -> 1 (depth 4)
  221 -> 12 -> 11 -> 2 -> 1 (depth 4)
  2111 -> 13 -> 11 -> 2 -> 1 (depth 4)
  11111 -> 5 -> 1 (depth 2)
Summary: 1 partition has depth 1; 1 has depth 2; 2 have 3; and 3 have 4, so that the row for n = 5 is 1..1..2..3 .

Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Clark Kimberling)

Row sums are A000041.
Column k = 2 is A032741.
Column k = 3 is A325245.
a(n!) = A325272(n).
Cf. A181819, A182850, A225486, A323014, A323023, A325238, A325239, A325254.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
t[s_] := t[s] = Length[f[s]]
u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], i]],
  {i, 1, Length[IntegerPartitions[n]]}];
Flatten[Table[Count[u[n], k], {n, 2, 25}, {k, 1, Max[u[n]]}]]

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.

A325272 Adjusted frequency depth of n!.

Original entry on oeis.org

0, 1, 3, 4, 5, 4, 6, 6, 6, 4, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 8, 7, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

			Recursively applying A181819 starting with 120 gives 120 -> 20 -> 6 -> 4 -> 3, so a(5) = 5.

a(n) = A001222(A325275(n)).
Cf. A000142, A006939, A303555, A323023, A325238, A325273, A325274, A325276, 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).

fd[n_]:=Switch[n,1,0,?PrimeQ,1,,1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
Table[fd[n!],{n,30}]

a(n) = A323014(n!).

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).

A325273 Prime omicron of n!.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).
The prime omicron of n (A304465) is 0 if n is 1, 1 if n is prime, and otherwise the second-to-last part of the omega-sequence of n. For example, the prime omicron of 180 is 2.
Conjecture: all terms after a(10) = 4 are less than 4.
From James Rayman, Apr 17 2021: (Start)
The conjecture is false. a(3804) = 4. In fact, there are 91 values of n < 10000 such that a(n) = 4.
The first value of n such that a(n) = 5 is 37934. For any other n < 5*10^5, a(n) < 5. (End)

James Rayman, Table of n, a(n) for n = 0..10000

a(n) = A055396(A325275(n)/2).
Cf. A000142, A006939, A303555, A323023, A325238, A325272, A325274, A325275, A325276, 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).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
omicron[n_]:=Switch[n,1,0,?PrimeQ,1,,omseq[n][[-2]]];
Table[omicron[n!],{n,0,100}]

from sympy.ntheory import *
def red(v):
    r = {}
    for i in v: r[i] = r.get(i, 0) + 1
    return r
def omicron(v):
    if len(v) == 0: return 0
    if len(v) == 1: return v[0]
    else: return omicron(list(red(v).values()))
f, a_list = {}, []
for i in range(101):
    a_list.append(omicron(list(f.values())))
    g = factorint(i+1)
    for k in g: f[k] = f.get(k, 0) + g[k]
print(a_list) # James Rayman, Apr 17 2021

More terms from James Rayman, Apr 17 2021

A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 8, 5, 9, 7, 10, 11, 12, 9, 7, 13, 14, 15, 16, 7, 17, 18, 14, 19, 20, 15, 21, 22, 23, 24, 15, 25, 13, 26, 27, 13, 28, 21, 29, 30, 31, 32, 11, 33, 34, 35, 36, 21, 37, 38, 39, 40, 25, 13, 41, 42, 43, 44, 33, 45, 35, 46, 47, 48, 21, 49, 19

Offset: 1

Gus Wiseman, May 25 2022

nonn

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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
This is the iteration of the transformation f described by Kimberling at A237685.

			Triangle begins:
   1
   2
   3
   4  3
   5
   6
   7
   8  5
   9  7
  10
  11
  12  9  7
Row 87780 is the following trajectory (left column), with prime indices shown on the right:
  87780: {1,1,2,3,4,5,8}
  65835: {2,2,3,4,5,8}
  51205: {3,4,4,5,8}
  19855: {3,5,8,8}
   2915: {3,5,16}

The version for run-lengths instead of sums is A325239 or A325277.
This is the iteration of A353832, with composition version A353847.
Row-lengths are A353841, counted by A353846.
Final terms are A353842.
Counting rows by final omega gives A353843.
Rows ending in a prime number are A353844, counted by A353845.
These sequences for compositions are A353853-A353859.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A182850 or A323014 gives frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353862 gives greatest run-sum of prime indices, least A353931.
Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.

Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]

cp's OEIS Frontend

A318283 Sum of elements of the multiset spanning an initial interval of positive integers with multiplicities equal to the prime indices of n in weakly decreasing order.

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A182857 Smallest number that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

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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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A304465 If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

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A225485 Number of partitions of n that have frequency depth k, an array read by rows.

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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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A325272 Adjusted frequency depth of n!.

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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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