A062770 -id:A062770 - cp's OEIS frontend

A071625 Number of distinct exponents when n is factorized as a product of primes.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Labos Elemer, May 29 2002

nonn

First term greater than 2 is a(360) = 3.
From Michel Marcus, Apr 24 2016: (Start)
A006939(n) gives the least m such that a(m) = n.
A062770 is the sequence of integers m such that a(m) = 1. (End)
We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first two omegas are A001222 and A001221, while this sequence is the third, and A323022 is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices). - Gus Wiseman, Jan 02 2019
Sanna (2020) proved that for each k>=1, the sequence of numbers n with A071625(n) = k has an asymptotic density A_k = (6/Pi^2) * Sum_{n>=1, n squarefree} rho_k(n)/psi(n), where psi is the Dedekind psi function (A001615), and rho_k(n) is defined by rho_1(n) = 1 if n = 1 and 0 otherwise, rho_{k+1}(n) = 0 if n = 1 and (1/(n-1)) * Sum_{d|n, dAmiram Eldar, Oct 18 2020

			n = 5040 = 2^4*(3*5)^2*7, three different exponents arise:4,2 and 1; so a(5040)=3.

Giovanni Resta, Table of n, a(n) for n = 1..10000
E. T. Bell, Functions of coprime divisors of integers, Bull. Amer. Math. Soc. 43 (1937), 818-822.
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link, arXiv preprint, arXiv:1902.09224 [math.NT], 2019.

Cf. A051903, A051904, A006939, A124010.

Cf. A001221, A001222, A001615, A059404, A062770, A118914, A181819, A323014, A323022, A323023, A323024, A323025.

# Using function 'PrimeSignature' from A124010.
a := n -> nops(convert(PrimeSignature(n), set)):
seq(a(n), n = 1..105); # Peter Luschny, Jun 15 2025

ffi[x_] := Flatten[FactorInteger[x]];
lf[x_] := Length[FactorInteger[x]];
ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}];
Table[Length[Union[ep[w]]], {w, 1, 256}]
(* Second program: *)
{0}~Join~Array[Length@ Union@ FactorInteger[#][[All, -1]] &, 104, 2] (* Michael De Vlieger, Apr 10 2019 *)

a(n) = #Set(factor(n)[,2]); \\ Michel Marcus, Mar 12 2015

from sympy import factorint
def a(n): return len(set(factorint(n).values()))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Sep 01 2022

A072774 Powers of squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Reinhard Zumkeller, Jul 10 2002

nonn

Essentially the same as A062770. - R. J. Mathar, Sep 25 2008
Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - Reinhard Zumkeller, Apr 06 2014
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A059404.
Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
A subsequence of A242414.
Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.

import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
   (1, 1, 1) : f (tail a005117_list) empty where
   f vs'@(v:vs) m
    | Map.null m || xx > v = (v, v, 1) :
                             f vs (insert (v^2) (v, 2) m)
    | otherwise = (xx, bx, ex) :
                  f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014

isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
select(isA, [seq(1..97)]);  # Peter Luschny, Jun 10 2025

Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)

is(n)=ispower(n,,&n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A072774(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

a(n) = A072775(n)^A072776(n).
Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025
a(n)/n ~ Pi^2/6 (A013661). - Friedjof Tellkamp, Jun 09 2025

A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

The indices of terms greater than 1 are {60, 84, 90, 120, 126, 132, 140, 150, ...}.
First term greater than 2 is a(1801800) = 3. In general, the first appearance of k is a(A182856(k)) = k.
The prime signature of n (row n of A118914) is the multiset of prime multiplicities in n.
We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first three omegas are A001222, A001221, A071625, and this sequence is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices).

			The prime signature of 1286485200 is {1, 1, 1, 2, 2, 3, 4}, in which 1 appears three times, two appears twice, and 3 and 4 both appear once, so there are 3 distinct multiplicities {1, 2, 3} and hence a(1286485200) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001221, A001222, A006939, A025487, A056239, A059404, A062770, A071625, A118914, A181819, A182856, A182857, A304464, A304465, A323014, A323023.

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
Table[PrimeNu[red[red[n]]],{n,200}]

a(n) = my(e=factor(n)[, 2], s = Set(e), m=Map(), v=vector(#s)); for(i=1, #s, mapput(m,s[i],i)); for(i=1, #e, v[mapget(m,e[i])]++); #Set(v) \\ David A. Corneth, Jan 02 2019

A071625(n) = #Set(factor(n)[, 2]); \\ From A071625
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A323022(n) = A071625(A181819(n)); \\ Antti Karttunen, Jan 03 2019

a(n) = A071625(A181819(n))
     = A001221(A181819(A181819(n)))
     = A001222(A181819(A181819(A181819(n))))
     = A056239(A181819(A181819(A181819(A181819(n))))).

More terms from Antti Karttunen, Jan 03 2019

A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

Original entry on oeis.org

1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 19, 15, 26, 1, 13, 39, 4, 25, 47, 5, 19, 70, 12, 29, 89, 17, 33, 115, 28, 42, 148, 41, 39, 189, 69, 62, 235, 88, 55, 294, 141, 81, 362, 183, 1, 84, 450, 253, 5, 103, 558, 333, 8, 105, 669, 464, 17, 153, 817, 576, 29

Offset: 1

Gus Wiseman, Apr 15 2019

For example, the partition (32111) has multiplicities {1,1,3}, of which 2 are distinct, so is counted under T(8,2).

			Triangle begins:
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  19
  15  26   1
  13  39   4
  25  47   5
  19  70  12
  29  89  17
  33 115  28
  42 148  41
  39 189  69
  62 235  88
  55 294 141
  81 362 183   1
Row n = 8 counts the following partitions:
  (8)         (332)
  (44)        (422)
  (53)        (611)
  (62)        (3221)
  (71)        (4211)
  (431)       (5111)
  (521)       (22211)
  (2222)      (32111)
  (3311)      (41111)
  (11111111)  (221111)
              (311111)
              (2111111)

Alois P. Heinz, Rows n = 1..200, flattened

Row lengths are A056556. Row sums are A000041. Column k = 1 is A047966. Column k = 2 is A325243.

Cf. A008284, A062770, A071625, A098859, A116608, A127002, A183558, A243978, A244515, A325244, A325268.

DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==k&]],{n,20},{k,n}],0,2]

A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Gus Wiseman, Nov 26 2023

nonn

As an operation on multisets, this is represented by A367579.

			90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so a(90) = 12.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
Agrees with A181819 at positions A367683, counted by A367682.
Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
Positions of first appearances are A367584, sorted A367585.
Positions of powers of 2 are A367586.
Divides n at positions A367685, counted by A367684.
The opposite version (cokernel) is A367859.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,100}]

a(n^k) = a(n) for all positive integers n and k.
A001221(a(n)) = A071625(n).
A001222(a(n)) = A001221(n).
If n is squarefree, a(n) = A020639(n)^A001222(n).
A056239(a(n)) = A367581(n).

A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 6, 1, 1, 2, 2, 1, 7, 1, 2, 8, 1, 3, 2, 2, 1, 1, 9, 1, 2, 3, 1, 1, 2, 1, 4, 10, 1, 1, 1, 11, 1, 2, 2, 1, 1, 3, 3, 1, 1, 12, 1, 1, 2, 2, 1, 3, 13, 1, 1, 1, 14, 1, 5, 2, 3, 1, 1, 15, 1, 2, 4, 1, 3, 2, 2, 1, 6, 16, 1, 2

Offset: 1

Gus Wiseman, Nov 25 2023

Row n = 1 is empty.
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.
We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.
Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.

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

Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Depends only on rootless base A052410, see A007916.
Row minima are A055396.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Rows have Heinz numbers A367580.
Row sums are A367581.
Row maxima are A367583, opposite A367587.
Index of first row with Heinz number n is A367584.
Sorted row indices of first appearances are A367585.
Indices of rows of the form {1,1,...} are A367586.
Agrees with sorted prime signature at A367683, counted by A367682.
A submultiset of prime indices at A367685, counted by A367684.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

For all positive integers n and k, row n^k is the same as row n.

A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

In each case, 2 is the fixed point that is reached (1 is the other fixed point of the x -> A181819(x) map).
Includes all integers whose prime signature a) contains two or more distinct numbers, and b) contains no number that occurs the same number of times as any other number. The first member of this sequence that does not fit that description is 75675600, whose prime signature is (4,3,2,2,1,1).
A full characterization is: Numbers whose prime signature (1) has not all equal multiplicities but (2) the numbers of distinct parts appearing with each distinct multiplicity are all equal. For example, the prime signature of 2520 is {1,1,2,3}, which satisfies (1) but fails (2), as the numbers of distinct parts appearing with each distinct multiplicity are 1 (with multiplicity 2, the part being 1) and 2 (with multiplicity 1, the parts being 2 and 3). Hence the sequence does not contain 2520. - Gus Wiseman, Jan 02 2019

			1. 180 requires exactly five iterations under the x -> A181819(x) map to reach a fixed point (namely, 2).  A181819(180) = 18;  A181819(18) = 6; A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).
2. The prime signature of 180 (2^2*3^2*5) is (2,2,1).
a. Two distinct numbers appear in (2,2,1) (namely, 1 and 2).
b. Neither 1 nor 2 appears in (2,2,1) the same number of times as any other number that appears there.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 5. See also A182853, A182854.
Subsequence of A059404 and A182851.  Includes A085987 and A179642 as subsequences.
Cf. A001221, A001222, A006939, A062770, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023.

Select[Range[1000],With[{sig=Sort[Last/@FactorInteger[#]]},And[!SameQ@@Length/@Split[sig],SameQ@@Length/@Union/@GatherBy[sig,Length[Position[sig,#]]&]]]&] (* Gus Wiseman, Jan 02 2019 *)

A323024 Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.

Original entry on oeis.org

360, 504, 540, 600, 720, 756, 792, 936, 1008, 1176, 1188, 1200, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1584, 1620, 1656, 1836, 1872, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2268, 2352, 2400, 2448, 2484, 2520, 2600, 2646, 2664, 2736, 2800, 2880, 2904

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

Positions of 3's in A071625.
Numbers k such that A001221(A181819(k)) = 3.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.030575..., where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d|n, 1Amiram Eldar, Oct 18 2020

			1500 = 2^2 * 3^1 * 5^3 has three distinct exponents {1, 2, 3}, so belongs to the sequence.
52500 = 2^2 * 3^1 * 5^4 * 7^1 has three distinct exponents {1, 2, 4}, so belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

Cf. A001221, A001222, A001615, A006939, A033992, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323025.

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[1000],tom[#]==3&]

is(n) = #Set(factor(n)[, 2]) == 3 \\ David A. Corneth, Jan 02 2019

A323025 Numbers with exactly four distinct exponents in their prime factorization, or four distinct parts in their prime signature.

Original entry on oeis.org

75600, 105840, 113400, 118800, 126000, 140400, 151200, 158760, 178200, 183600, 198000, 205200, 210600, 211680, 232848, 234000, 237600, 246960, 248400, 252000, 261360, 275184, 275400, 280800, 283500, 294000, 302400, 306000, 307800, 313200, 315000, 334800

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

Positions of 4's in A071625.
Numbers k such that A001221(A181819(k)) = 4.
Is a(n) ~ c * n for some c? - David A. Corneth, Jan 09 2019
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.00035750... (corresponding to c = 2797.1... in the question above, whose answer is affirmative), where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d_1|n, 1Amiram Eldar, Oct 18 2020

			126000 = 2^4 * 3^2 * 5^3 * 7^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.
831600 = 2^4 * 3^3 * 5^2 * 7^1 * 11^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

Cf. A001221, A001222, A001615, A006939, A033993, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323024.

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[100000],tom[#]==4&]

is(n) = #Set(factor(n)[, 2]) == 4 \\ David A. Corneth, Jan 09 2019

A323055 Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(2) = 12.
First differs from A059404 in lacking 360, whose prime signature has three distinct parts.
Positions of 2's in A071625.
Numbers k such that A001221(A181819(k)) = 2.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} 1/((n-1)*psi(n)) = 0.3611398..., where psi is the Dedekind psi function (A001615) (Sanna, 2020). - Amiram Eldar, Oct 18 2020

			3000 = 2^3 * 3^1 * 5^3 has two distinct exponents {1, 3}, so belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

One distinct exponent: A062770 or A072774.
Two distinct exponents: this sequence.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A001615, A006939, A007774, A059404, A071625, A118914, A181819, A323014, A323022.

isA323055 := proc(n)
    local eset;
    eset := {};
    for pf in ifactors(n)[2] do
        eset := eset union {pf[2]} ;
    end do:
    simplify(nops(eset) = 2 ) ;
end proc:
for n from 12 to 1000 do
    if isA323055(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 09 2019

Select[Range[100],Length[Union[Last/@FactorInteger[#]]]==2&]

cp's OEIS Frontend

A071625 Number of distinct exponents when n is factorized as a product of primes.

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A072774 Powers of squarefree numbers.

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A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n.

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A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

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A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

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A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

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