A318284 -id:A318284 - cp's OEIS frontend

A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 9, 3, 1, 1, 15, 66, 31, 5, 1, 1, 52, 712, 686, 109, 7, 1, 1, 203, 10457, 27036, 6721, 339, 11, 1, 1, 877, 198091, 1688360, 911838, 58616, 1043, 15, 1, 1, 4140, 4659138, 154703688, 231575143, 26908756, 476781, 2998, 22, 1

Offset: 0

Alois P. Heinz, Nov 26 2012

A(n,k) is the number of factorizations of m^n where m is a product of k distinct primes. A(2,2) = 9: (2*3)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.

A(n,k) is the number of (n*k) X k matrices with nonnegative integer entries and column sums n up to permutation of rows. - Andrew Howroyd, Dec 10 2018

			A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
A(2,2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].
Square array A(n,k) begins:
  1,   1,    1,      1,        1,         1,         1,       1, ...
  1,   1,    2,      5,       15,        52,       203,     877, ...
  1,   2,    9,     66,      712,     10457,    198091, 4659138, ...
  1,   3,   31,    686,    27036,   1688360, 154703688, ...
  1,   5,  109,   6721,   911838, 231575143, ...
  1,   7,  339,  58616, 26908756, ...
  1,  11, 1043, 476781, ...
  1,  15, 2998, ...

Andrew Howroyd, Table of n, a(n) for n = 0..209

Columns k=0..3 give: A000012, A000041, A002774, A219678.
Rows n=0..4 give: A000012, A000110, A020555, A322487, A358781.
Main diagonal gives A322488.
Cf. A188392, A219585 (partitions of {n}^k into distinct k-tuples), A256384, A318284, A318951.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 11 2018

A076716 Number of distinct factorizations of n! with all factors > 1.

Original entry on oeis.org

1, 1, 2, 7, 21, 98, 392, 2116, 11830, 70520, 425240, 2787810, 19530213, 144890639, 1149978830, 8558078111, 76417516719, 618437486332, 6087770992601, 54574732902278, 525656554130914, 5290117056157616, 61626071051832409, 555057889968635744, 5809502058957961682

Offset: 1

Donald S. McDonald, Oct 27 2002

nonn

			a(3) = 2 because 3! = 6 = 2*3 has just 2 factorizations.
a(4) = 7 because 4! = 24 = 2*12 = 2*2*6 = 2*2*2*3 = 2*3*4 = 3*8 = 4*6 has 7 factorizations.

Cf. A001055, A157612, A162247, A318284.

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-> b(n!$2):
seq(a(n), n=1..12);  # Alois P. Heinz, May 25 2013

c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r]=Module[{ds, i}, ds=Select[Divisors[n], 1<#<=r&]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n!, n! ]; a/@Range[16] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson, Oct 29 2002 *)

\\ See A318284 for count.
a(n)={if(n<=1, 1, count(factor(n!)[,2]))} \\ Andrew Howroyd, Feb 01 2020

a(n) = A001055(n!).

Edited by Robert G. Wilson v, Oct 29 2002
4 more terms from Ryan Propper, May 20 2007
a(20)-a(25) from Andrew Howroyd, Feb 01 2020

A334030 Number of combinatory separations 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, 3, 5, 7, 8, 8, 7, 9, 8, 5, 7, 12, 15, 14, 15, 17, 18, 13, 12, 17, 17, 16, 14, 16, 13, 7, 11, 19, 27, 26, 27, 37, 37, 25, 27, 37, 33, 34, 37, 40, 36, 22, 19, 32, 37, 33, 37, 38, 40, 28, 26, 33, 34, 30, 25, 28, 22, 11, 15, 30, 44, 42, 51, 68

Offset: 0

Gus Wiseman, Apr 16 2020

nonn

A multiset is normal if it covers an initial interval of positive integers. The type of a multiset of integers 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 (3,3,5,5,5,6) is (1,1,2,2,2,3).

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset (1122) are (1122), (1)(112), (1)(122), (11)(11), (12)(12), (1)(1)(11), (1)(1)(12), (1)(1)(1)(1). This list excludes (12)(11), because one cannot partition (1122) into two blocks where one block has two distinct elements and the other has two equal elements.

			The combinatory separations for n = 1, 3, 5, 9, 10, 13 (heads not shown):
  (1)  (12)    (112)      (1112)        (1122)        (1223)
       (1)(1)  (1)(11)    (1)(111)      (11)(11)      (1)(112)
               (1)(12)    (1)(112)      (1)(112)      (11)(12)
               (1)(1)(1)  (11)(12)      (1)(122)      (1)(122)
                          (1)(1)(11)    (12)(12)      (1)(123)
                          (1)(1)(12)    (1)(1)(11)    (12)(12)
                          (1)(1)(1)(1)  (1)(1)(12)    (1)(1)(11)
                                        (1)(1)(1)(1)  (1)(1)(12)
                                                      (1)(1)(1)(1)

Multisets of compositions are A034691.
The described multiset is a row of A095684.
Combinatory separations of normal multisets are A269134.
Shuffles of compositions are counted by A292884.
Combinatory separations of prime indices are A318559.
The version for prime indices is A318560.
Combinatory separations of strongly normal multisets are A318563.
Multiset partitions of the described multiset are A333942.
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.
Cf. A065609, A275692, A318284, A326774, A333764, A333765, A333940.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
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]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
Table[Length[Union[Table[Sort[normize/@m],{m,mps[ptnToNorm[stc[n]]]}]]],{n,0,100}]

A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 7, 7, 7, 9, 11, 12, 16, 5, 15, 17, 22, 16, 29, 19, 30, 16, 21, 30, 23, 29, 42, 52, 56, 7, 47, 45, 57, 43, 77, 67, 77, 31, 101, 98, 135, 47, 85, 97, 176, 29, 66, 64, 118, 77, 231, 69, 97, 57, 181, 139, 297, 137, 385, 195, 166, 11, 162, 171, 490, 118

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

			Non-isomorphic representatives of the a(12) = 9 multiset partitions of {1,1,2,3}:
  {{1,1,2,3}}
  {{1},{1,2,3}}
  {{2},{1,1,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..130

Cf. A007716, A181821, A255906, A269134, A305936, A317533, A317791.

Cf. A318283, A318284, A318287, A318362, A318371.

\\ See links in A339645 for combinatorial species functions.
sig(n)={my(f=factor(n), sig=vector(primepi(vecmax(f[,1])))); for(i=1, #f~, sig[primepi(f[i,1])]=f[i,2]); sig}
C(sig)={my(n=sum(i=1, #sig, i*sig[i]), A=Vec(symGroupSeries(n)-1), B=O(x*x^n), c=prod(i=1, #sig, if(sig[i], sApplyCI(A[sig[i]], sig[i], A[i], i), 1))); polcoef(OgfSeries(sCartProd(c*x^n + B, sExp(x*Ser(A) + B))), n)}
a(n)={if(n==1, 1, C(sig(n)))} \\ Andrew Howroyd, Jan 17 2023

a(n) = A317791(A181821(n)).

Terms a(31) and beyond from Andrew Howroyd, Jan 17 2023

A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 25 2004

Row k is the unique multiset that covers an initial interval of positive integers and has multiplicities equal to the parts of the k-th composition in standard order (graded reverse-lexicographic, A066099). This composition 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. For example, the 13th composition is (1,2,1), so row 13 is {1,2,2,3}. - Gus Wiseman, Apr 26 2020

			1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, ...
The 8 strings of length 4 are 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234.
From _Gus Wiseman_, Apr 26 2020: (Start)
The triangle read by columns begins:
  1:{1}  2:{1,1}  4:{1,1,1}   8:{1,1,1,1}  16:{1,1,1,1,1}
         3:{1,2}  5:{1,1,2}   9:{1,1,1,2}  17:{1,1,1,1,2}
                  6:{1,2,2}  10:{1,1,2,2}  18:{1,1,1,2,2}
                  7:{1,2,3}  11:{1,1,2,3}  19:{1,1,1,2,3}
                             12:{1,2,2,2}  20:{1,1,2,2,2}
                             13:{1,2,2,3}  21:{1,1,2,2,3}
                             14:{1,2,3,3}  22:{1,1,2,3,3}
                             15:{1,2,3,4}  23:{1,1,2,3,4}
                                           24:{1,2,2,2,2}
                                           25:{1,2,2,2,3}
                                           26:{1,2,2,3,3}
                                           27:{1,2,2,3,4}
                                           28:{1,2,3,3,3}
                                           29:{1,2,3,3,4}
                                           30:{1,2,3,4,4}
                                           31:{1,2,3,4,5}
(End)

J. C. Kieffer, W. Szpankowski and E.-H. Yang, Problems on sequences: information theory and computer science interface, IEEE Trans. Inform. Theory, 50 (No. 7, 2004), 1385-1392.

See A096299 for another version.
The number of distinct parts in row n is A000120(n), also the maximum part.
Row sums are A029931.
Heinz numbers of rows are A057335.
Row lengths are A070939.
Row products are A284001.
The version for prime indices is A305936.
There are A333942(n) multiset partitions of row n.
Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are A269134.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Dealings are counted by A333939.
- Distinct parts are counted by A334028.
Cf. A318284, A333764, A333940, A334030.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
Table[ptnToNorm[stc[n]],{n,15}] (* Gus Wiseman, Apr 26 2020 *)

A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 3, 12, 18, 12, 2, 36, 4, 10, 20, 72, 2, 60, 4, 40, 60, 24, 3, 120, 80, 14, 360, 120, 4, 240, 2, 240, 42, 32, 70, 720, 6, 27, 112, 480, 2, 210, 4, 84, 420, 40, 4, 1440, 280, 280, 108, 224, 5, 1260, 224, 420, 180, 22, 2, 840, 6, 72, 1680, 2880

Offset: 1

Gus Wiseman, Jan 13 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(22) = 24 matrices:
  [111112] [111121] [111211] [112111] [121111] [211111]
.
  [111] [111] [111] [112] [121] [211]
  [112] [121] [211] [111] [111] [111]
.
  [11] [11] [11] [11] [12] [21]
  [11] [11] [12] [21] [11] [11]
  [12] [21] [11] [11] [11] [11]
.
  [1] [1] [1] [1] [1] [2]
  [1] [1] [1] [1] [2] [1]
  [1] [1] [1] [2] [1] [1]
  [1] [1] [2] [1] [1] [1]
  [1] [2] [1] [1] [1] [1]
  [2] [1] [1] [1] [1] [1]

Gus Wiseman, The a(27) = 360 matrix-arrangements of (1,1,2,2,3,3).

Cf. A007716, A056239, A112798, A120733, A181821, A305936, A318283, A318284, A323295, A323300, A323351.

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]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Array[Length[ptnmats[Times@@Prime/@nrmptn[#]]]&,30]

a(n) = A318762(n) * A000005(A056239(n)).

A330989 Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.

Original entry on oeis.org

1, 4, 12, 0, 72, 0, 480

Offset: 0

Gus Wiseman, Jan 07 2020

			The A001055(n) factorizations for n = 1, 4, 12, 72:
  ()  (4)    (12)     (72)
      (2*2)  (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)

All nonzero terms belong to A025487 and also A033833.
Factorizations are A001055, with image A045782.
The least number with exactly n factorizations is A330973(n).
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly prime(n) factorizations is A330992(n).
Cf. A002033, A045778, A045783, A318284, A330935, A330972, A330976, A330990, A330991, A331022.

A284001 a(n) = A005361(A283477(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 6, 1, 2, 4, 6, 8, 12, 18, 24, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480, 600, 720, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480

Offset: 0

Antti Karttunen, Mar 18 2017

a(n) is the product of elements of the multiset that covers an initial interval of positive integers with multiplicities equal to the parts of the n-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1) giving the multiset {1,2,2,3} with product 12, so a(13) = 12. - Gus Wiseman, Apr 26 2020

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A005361, A283477, A284002, A284005.
Row products of A095684.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Weighted sum is A029931.
- Necklaces are A065609.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Distinct parts are counted by A334028.
Cf. A003963, A034691, A053645, A057335, A305936, A318284, A333764, A333940, A333942, A334030.

Table[Times @@ FactorInteger[#][[All, -1]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 93}] (* Michael De Vlieger, Mar 18 2017 *)

A005361(n) = factorback(factor(n)[, 2]); \\ From A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A283477(n) = A108951(A019565(n));
A284001(n) = A005361(A283477(n));

(define (A284001 n) (A005361 (A283477 n)))

a(n) = A005361(A283477(n)).
a(n) = A003963(A057335(n)). - Gus Wiseman, Apr 26 2020
a(n) = A284005(A053645(n)) for n > 0 with a(0) = 1. - Mikhail Kurkov, Jun 05 2021 [verification needed]

A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).

Is a(k) = A000110(k) for A025487(m) = A002110(k)?

			From _Gus Wiseman_, Jan 13 2020: (Start)
The a(1) = 1 through a(11) = 9 factorizations:
  {}  2  4    6    8      12     16       24       30     32         36
         2*2  2*3  2*4    2*6    2*8      3*8      5*6    4*8        4*9
                   2*2*2  3*4    4*4      4*6      2*15   2*16       6*6
                          2*2*3  2*2*4    2*12     3*10   2*2*8      2*18
                                 2*2*2*2  2*2*6    2*3*5  2*4*4      3*12
                                          2*3*4           2*2*2*4    2*2*9
                                          2*2*2*3         2*2*2*2*2  2*3*6
                                                                     3*3*4
                                                                     2*2*3*3
(End)

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..5000 (First 300 terms from _R. J. Mathar_)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Cf. A000041, A000079, A000110, A001055, A002110, A025487.
The version indexed by unsorted prime signature is A331049.
The version indexed by prime shadow (A181819, A181821) is A318284.
This sequence has range A045782 (same as A001055).
Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

A050322 := proc(n)
    A001055(A025487(n)) ;
end proc: # R. J. Mathar, May 25 2017

c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length/@facs/@First/@GatherBy[Range[1000],If[#==1,{},Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)

A330998 Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

This is the sorted list of positions of first appearances in A318284 of each element of the range A045782.

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.

			Factorizations of the inverse prime shadows of the initial terms:
    4    8      12     16       36       24       60       48
    2*2  2*4    2*6    2*8      4*9      3*8      2*30     6*8
         2*2*2  3*4    4*4      6*6      4*6      3*20     2*24
                2*2*3  2*2*4    2*18     2*12     4*15     3*16
                       2*2*2*2  3*12     2*2*6    5*12     4*12
                                2*2*9    2*3*4    6*10     2*3*8
                                2*3*6    2*2*2*3  2*5*6    2*4*6
                                3*3*4             3*4*5    3*4*4
                                2*2*3*3           2*2*15   2*2*12
                                                  2*3*10   2*2*2*6
                                                  2*2*3*5  2*2*3*4
                                                           2*2*2*2*3
The corresponding multiset partitions:
    {11}    {111}      {112}      {1111}        {1122}        {1112}
    {1}{1}  {1}{11}    {1}{12}    {1}{111}      {1}{122}      {1}{112}
            {1}{1}{1}  {2}{11}    {11}{11}      {11}{22}      {11}{12}
                       {1}{1}{2}  {1}{1}{11}    {12}{12}      {2}{111}
                                  {1}{1}{1}{1}  {2}{112}      {1}{1}{12}
                                                {1}{1}{22}    {1}{2}{11}
                                                {1}{2}{12}    {1}{1}{1}{2}
                                                {2}{2}{11}
                                                {1}{1}{2}{2}

Taking n instead of the inverse prime shadow of n gives A330972.
Factorizations are A001055, with image A045782, with complement A330976.
Factorizations of inverse prime shadows are A318284.
Cf. A025487, A033833, A045778, A045783, A181819, A181821, A318286, A325238, A330973, A330989, A330990, A330993, A330997.

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}]]]]];
nds=Table[Length[facs[Times@@Prime/@nrmptn[n]]],{n,50}];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

cp's OEIS Frontend

A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A076716 Number of distinct factorizations of n! with all factors > 1.

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A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.

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A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

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A330989 Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.

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A284001 a(n) = A005361(A283477(n)).

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