A181796 -id:A181796 - cp's OEIS frontend

A336866 Number of integer partitions of n without all distinct multiplicities.

0, 0, 0, 1, 1, 2, 4, 5, 9, 15, 21, 28, 46, 56, 80, 114, 149, 192, 269, 337, 455, 584, 751, 943, 1234, 1527, 1944, 2422, 3042, 3739, 4699, 5722, 7100, 8668, 10634, 12880, 15790, 19012, 23093, 27776, 33528, 40102, 48264, 57469, 68793, 81727, 97372, 115227

Offset: 0

Gus Wiseman, Aug 09 2020

nonn

			The a(0) = 0 through a(9) = 15 partitions (empty columns shown as dots):
  .  .  .  (21)  (31)  (32)  (42)    (43)    (53)     (54)
                       (41)  (51)    (52)    (62)     (63)
                             (321)   (61)    (71)     (72)
                             (2211)  (421)   (431)    (81)
                                     (3211)  (521)    (432)
                                             (3221)   (531)
                                             (3311)   (621)
                                             (4211)   (3321)
                                             (32111)  (4221)
                                                      (4311)
                                                      (5211)
                                                      (32211)
                                                      (42111)
                                                      (222111)
                                                      (321111)

A098859 counts the complement.
A130092 gives the Heinz numbers of these partitions.
A001222 counts prime factors with multiplicity.
A013929 lists nonsquarefree numbers.
A047966 counts uniform partitions.
A047967 counts non-strict partitions.
A071625 counts distinct prime multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor with distinct prime multiplicities.
Cf. A000009, A000041, A008284, A116608, A118914, A124010, A242882, A255231, A325280, A325242, A329739, A336422, A336571.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Length/@Split[#]&]],{n,0,30}]

a(n) = A000041(n) - A098859(n).

A327498 Maximum divisor of n whose prime multiplicities are distinct (A130091).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 7, 5, 16, 17, 18, 19, 20, 7, 11, 23, 24, 25, 13, 27, 28, 29, 5, 31, 32, 11, 17, 7, 18, 37, 19, 13, 40, 41, 7, 43, 44, 45, 23, 47, 48, 49, 50, 17, 52, 53, 54, 11, 56, 19, 29, 59, 20, 61, 31, 63, 64, 13, 11, 67, 68, 23

Offset: 1

Gus Wiseman, Sep 16 2019

A number's prime multiplicities are also called its (unsorted) prime signature.

Every positive integer appears a finite number of times in the sequence; a prime p occurs 2^(PrimePi(p) - 1) times. - David A. Corneth, Sep 17 2019

			The divisors of 60 whose prime multiplicities are distinct are {1, 2, 3, 4, 5, 12, 20}, so a(60) = 20, the largest of these divisors.

David A. Corneth, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A000720, A007916, A098859, A124010, A181796, A212168, A255231, A327499, A351564.

Table[Max[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&]],{n,100}]

a(n) = {my(m = Map(), f = factor(n), res = 1); forstep(i = #f~, 1, -1, forstep(j = f[i, 2], 1, -1, if(!mapisdefined(m, j), mapput(m, j, j); res*=f[i, 1]^j; next(2)))); res} \\ David A. Corneth, Sep 17 2019

A351564(n) = issquarefree(factorback(apply(e->prime(e),(factor(n)[,2]))));
A327498(n) = fordiv(n,d,if(A351564(n/d), return(n/d))); \\ Antti Karttunen, Apr 02 2022

a(A130091(n)) = n and a(A130092(n)) < n. - Ivan N. Ianakiev, Sep 17 2019

a(n) = n / A327499(n). - Antti Karttunen, Apr 02 2022

A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 5, 5, 5, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 7, 8, 9, 3, 5, 5, 8, 4, 8, 7, 11, 5, 8, 7, 11, 7, 11, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 5, 7, 8, 9, 3, 5, 5, 8, 5, 7

Offset: 0

Gus Wiseman, Jun 21 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, 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.

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(180) = 7 patterns are: (), (1), (1,2), (2,1), (1,2,3), (2,1,2), (2,1,2,3).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The non-contiguous version is A335454.
Summing over indices with binary length n gives A335457(n).
The nonempty version is A335474.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A108917, A124767, A124770, A181796, A269134, A333218, A333222, A333628, A334030, A335456.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,30}]

a(n) = A335474(n) + 1.

A335457 Number of normal patterns contiguously matched by compositions of n.

Original entry on oeis.org

1, 2, 5, 12, 31, 80, 196, 486, 1171, 2787, 6564, 15323, 35403, 81251, 185087, 418918, 942525, 2109143, 4695648, 10405694, 22959156

Offset: 0

Gus Wiseman, Jun 23 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(0) = 1 through a(3) = 12 pairs of a composition with a contiguously matched pattern:
  ()()  (1)()   (2)()     (3)()
        (1)(1)  (11)()    (12)()
                (2)(1)    (21)()
                (11)(1)   (3)(1)
                (11)(11)  (111)()
                          (12)(1)
                          (21)(1)
                          (111)(1)
                          (12)(12)
                          (21)(21)
                          (111)(11)
                          (111)(111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for standard compositions is A335458.
The non-contiguous version is A335456.
Patterns are counted by A000670 and ranked by A333217.
The n-th standard composition has A124771(n) contiguous subsequences.
Patterns contiguously matched by prime indices are A335549.
Minimal avoided patterns of prime indices are counted by A335550.
Cf. A000005, A056986, A108917, A124767, A181796, A269134, A333224, A334299.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[cmp,{_,s___,_}:>{s}]]],{cmp,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(16)-a(20) from Jinyuan Wang, Jul 08 2020

A336416 Number of perfect-power divisors of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 7, 7, 11, 18, 36, 36, 47, 47, 84, 122, 166, 166, 221, 221, 346, 416, 717, 717, 1001, 1360, 2513, 2942, 4652, 4652, 5675, 5675, 6507, 6980, 13892, 17212, 20408, 20408, 39869, 45329, 51018, 51018, 68758, 68758, 105573, 138617, 284718, 284718, 338126, 421126

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

			The a(1) = 0 through a(9) = 18 divisors:
       1: 1
       2: 1
       6: 1
      24: 1,4,8
     120: 1,4,8
     720: 1,4,8,9,16,36,144
    5040: 1,4,8,9,16,36,144
   40320: 1,4,8,9,16,32,36,64,128,144,576
  362880: 1,4,8,9,16,27,32,36,64,81,128,144,216,324,576,1296,1728,5184

David A. Corneth, Table of n, a(n) for n = 0..9999

The maximum among these divisors is A090630, with quotient A251753.
The version for distinct prime exponents is A336414.
The uniform version is A336415.
Replacing factorials with Chernoff numbers (A006939) gives A336417.
Prime powers are A000961.
Perfect powers are A001597, with complement A007916.
Prime power divisors are counted by A022559.
Cf. A000005, A001221, A001222, A011371, A032741, A118914, A124010, A130091, A327527.
Factorial numbers: A000142, A007489, A027423, A048656, A071626, A181796, A325272, A325273, A325617.

perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[n!],perpouQ]],{n,0,15}]

a(n) = sumdiv(n!, d, (d==1) || ispower(d)); \\ Michel Marcus, Aug 19 2020

addhelp(val, "exponent of prime p in n!")
val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = {if(n<=3, return(1)); my(pr = primes(primepi(n\2)), v = vector(#pr, i, val(n, pr[i])), res = 1, cv); for(i = 2, v[1], if(issquarefree(i), cv = v\i; res-=(prod(i = 1, #cv, cv[i]+1)-1)*(-1)^omega(i) ) ); res } \\ David A. Corneth, Aug 19 2020

a(p) = a(p-1) for prime p. - David A. Corneth, Aug 19 2020

a(26)-a(34) from Jinyuan Wang, Aug 19 2020

a(35)-a(49) from David A. Corneth, Aug 19 2020

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
  2  4    8      5*12     16       4*9      32         48
     2*2  2*4    3*20     4*4      3*12     4*8        4*12
          2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
                 2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
                          2*2*2*2  2*2*9    2*2*8      2*24
                                   2*2*3*3  2*2*2*4    2*3*8
                                            2*2*2*2*2  2*2*12
                                                       2*2*3*4
                                                       2*2*2*2*3

A327523 is the case when n is restricted to belong to A130091 also.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty chains of divisors.
A281116 counts factorizations with no common divisor.
A302696 lists numbers whose prime indices are pairwise coprime.
A305149 counts stable factorizations.
A320439 counts factorizations using A289509.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A336568 = not a product of two numbers with distinct prime multiplicities.
A336569 counts maximal chains of elements of A130091.
A337256 counts chains of divisors.
Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.

facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]

A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 2, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Mar 11 2020

nonn

Number of ordered prime factorizations of radical of n.

Number of permutations of the prime indices of n (counting multiplicity) avoiding the patterns (1,2,1) and (2,1,2). These are permutations with all equal parts contiguous. Depends only on sorted prime signature (A118914). - Gus Wiseman, Jun 27 2020

			From _Gus Wiseman_, Jun 27 2020 (Start)
The a(n) permutations of prime indices for n = 2, 12, 60:
  (1)  (112)  (1123)
       (211)  (1132)
              (2113)
              (2311)
              (3112)
              (3211)
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.
Dominates A335451.
Permutations of prime indices are A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1)-avoiding permutations of prime indices are A335449.
(2,1,2)-avoiding permutations of prime indices are A335450.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Cf. A056239, A112798, A181796, A333221, A335452, A335463, A335521.

f:= n -> nops(numtheory:-factorset(n))!:
map(f, [$1..100]); # Robert Israel, Mar 12 2020

a[1] = 1; a[n_] := a[n] = Plus @@ (a[n/#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n/d, d] == 1 && d < n, Boole[PrimePowerQ[n/d]] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
Table[PrimeNu[n]!, {n, 1, 100}]

a(1) = 1; a(n) = Sum_{d|n, d < n, gcd(d, n/d) = 1} A069513(n/d) * a(d).

a(n) = A000142(A001221(n)).

A336414 Number of divisors of n! with distinct prime multiplicities.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 20, 27, 48, 86, 147, 195, 311, 390, 595, 1031, 1459, 1791, 2637, 3134, 4747, 7312, 10766, 12633, 16785, 26377, 36142, 48931, 71144, 82591, 112308, 128023, 155523, 231049, 304326, 459203, 568095, 642446, 812245, 1137063, 1441067, 1612998, 2193307, 2429362

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number has distinct prime multiplicities iff its prime signature is strict.

			The first and second columns below are the a(6) = 20 counted divisors of 6! together with their prime signatures. The third column shows the A000005(6!) - a(6) = 10 remaining divisors.
      1: ()      20: (2,1)    |    6: (1,1)
      2: (1)     24: (3,1)    |   10: (1,1)
      3: (1)     40: (3,1)    |   15: (1,1)
      4: (2)     45: (2,1)    |   30: (1,1,1)
      5: (1)     48: (4,1)    |   36: (2,2)
      8: (3)     72: (3,2)    |   60: (2,1,1)
      9: (2)     80: (4,1)    |   90: (1,2,1)
     12: (2,1)  144: (4,2)    |  120: (3,1,1)
     16: (4)    360: (3,2,1)  |  180: (2,2,1)
     18: (1,2)  720: (4,2,1)  |  240: (4,1,1)

Chai Wah Wu, Table of n, a(n) for n = 0..6245 (n = 0..94 from David A. Corneth)

Perfect-powers are A001597, with complement A007916.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
The maximum divisor with distinct prime multiplicities is A327498.
Divisors of n! with equal prime multiplicities are counted by A336415.
Cf. A000005, A098859, A118914, A124010, A327527, A336424, A336500, A336568.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); #vecsort(ex,,8) == #ex); \\ Michel Marcus, Jul 24 2020

a(n) = A181796(n!).

a(21)-a(41) from Alois P. Heinz, Jul 24 2020

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(1) = 1 through a(16) = 5 divisors:
  1  1  1  1  1  2  1  1  1  2  1  1  1  2  3  1
     2  3  2  5  3  7  2  3  5 11  3 13  7  5  2
           4           4  9        4           4
                       8          12           8
                                              16

A336419 is the version for superprimorials.
A336568 gives positions of zeros.
A336869 is the restriction to factorials.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336568 gives numbers not a product of two elements of A130091.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A000005, A001055, A002033, A098859, A124010, A167865, A253249, A336870.

Table[Length[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n/#]&]],{n,25}]

cp's OEIS Frontend

A336866 Number of integer partitions of n without all distinct multiplicities.

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A327498 Maximum divisor of n whose prime multiplicities are distinct (A130091).

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A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

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A335457 Number of normal patterns contiguously matched by compositions of n.

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A336416 Number of perfect-power divisors of n!.

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A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

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A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

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