A118914 -id:A118914 - cp's OEIS frontend

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

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.

A064839 List the natural numbers starting a new row only with each new least prime signature (A025487). a(n) is the column position associated with n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 3, 4, 1, 7, 2, 8, 3, 5, 6, 9, 1, 3, 7, 2, 4, 10, 1, 11, 1, 8, 9, 10, 1, 12, 11, 12, 2, 13, 2, 14, 5, 6, 13, 15, 1, 4, 7, 14, 8, 16, 3, 15, 4, 16, 17, 17, 1, 18, 18, 9, 1, 19, 3, 19, 10, 20, 4, 20, 1, 21, 21, 11, 12, 22, 5, 22, 2, 2, 23, 23, 2, 24, 25, 26

Offset: 1

Alford Arnold, Oct 24 2001

Row 2 records the primes (A000040). Rows 3 and 4 record the semiprimes (A001358). Rows 5, 6 and 9 record the 3-almost primes (A014612) etc. A058933 is a similar sequence based on k-almost primes.
The graph of this sequence is interesting for large n because it shows multiple curves, one for each prime signature. For example, the six highest curves on the graph of a(n) for n up to 10^4 are for the (1,1), (1,1,1), (1), (2,1,1), (2,1), and (1,1,1,1) prime signatures. The (1) curve dominates until n=58; the (1,1) curve dominates until n=1279786, when the (1,1,1) curve intersects the (1,1) curve. Each (1,1,...,1) curve dominates for a finite number of n.
Ordinal transform of A101296. - Antti Karttunen, May 15 2017
a(n) is the number of positive integers up to n with the same prime signature as n. For example, the a(20) = 3 numbers are {12, 18, 20}. - Gus Wiseman, Jul 08 2019
Ordinal transform of A046523. - Alois P. Heinz, May 31 2020

			The list begins as follows:
1
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ...
4 9 25 49 ...
6 10 14 15 21 22 26 33 34 35 38 39 46 51 ...
8 27 ...
12 18 20 28 44 45 50 52 ...
16 ...
Note: the above array, without the initial 1, is given by A095904 (and its transpose A179216). - _Antti Karttunen_, May 15 2017

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A014612, A025487, A046523, A058933, A101296.
Cf. also arrays A095904, A179216.
Cf. A001222, A085089, A118914, A124010, A325263, A325365, A326438, A326439.

p:= proc() 0 end:
a:= proc(n) option remember; local t; a(n-1);
      t:= (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
           sort(map(i-> i[2], ifactors(n)[2]), `>`));
      p(t):= p(t)+1
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, May 31 2020

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Count[Array[prisig,n],prisig[n]],{n,30}] (* Gus Wiseman, Jul 08 2019 *)

More terms from Naohiro Nomoto, Oct 31 2001

A329140 Numbers whose prime signature is a periodic word.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A182853 in having 2100 = 2^2 * 3^1 * 5^2 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their prime signatures begins:
   6: (1,1)
  10: (1,1)
  14: (1,1)
  15: (1,1)
  21: (1,1)
  22: (1,1)
  26: (1,1)
  30: (1,1,1)
  33: (1,1)
  34: (1,1)
  35: (1,1)
  36: (2,2)
  38: (1,1)
  39: (1,1)
  42: (1,1,1)
  46: (1,1)
  51: (1,1)
  55: (1,1)
  57: (1,1)
  58: (1,1)

Complement of A329139.
Periodic compositions are A178472.
Periodic binary words are A152061.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A329132 A329134, A329143, A329144.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],!aperQ[Last/@FactorInteger[#]]&]

A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227, 229, 233, 239, 241, 251

Offset: 1

Gus Wiseman, May 15 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   36: {1,1,2,2}

Robert Israel, Table of n, a(n) for n = 1..10000

The first term that is not a prime power A000961 is 36.
The first term that is not a prime or a perfect power A001597 is 1260. - Corrected by Robert Israel, Mar 10 2025
The non-recursive version is A325755, counted by A325702.
Removing all primes gives A353389.
These partitions are counted by A353426.
The version for compositions is A353431.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with all distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

pshadow:= proc(n) local F,i;
  F:= ifactors(n)[2];
  mul(ithprime(i),i=F[..,2])
end proc:
filter:= proc(n) local s;
  if isprime(n) then return true fi;
  s:= pshadow(n);
  n mod s = 0 and member(s,R)
end proc:
R:= {}:
for i from 2 to 2000 do if filter(i) then R:= R union {i} fi od:
sort(convert(R,list)); # Robert Israel, Mar 10 2025

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,200],suQ[#]&]

Equals A353389 U A000040.

A353394 Product of prime shadows of prime indices of n (with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset: 1

Gus Wiseman, May 17 2022

nonn

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 prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.

Positions of first appearances are A353397.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, with an inverse A181821.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Times@@red/@primeMS[n],{n,100}]

a(n) = Product_i A181819(A112798(n,i)).
Positions where a(n) = A003963(n) are A003586.
Positions where a(n) = A005361(n) are A353399, counted by A353398.
Positions where a(n) = A181819(n) are A353395, counted by A353396.

A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

Original entry on oeis.org

1, 2, 12, 20, 36, 44, 56, 68, 100, 124, 164, 184, 208, 236, 240, 268, 332, 436, 464, 484, 508, 528, 608, 628, 688, 716, 720, 752, 764, 776, 816, 844, 880, 964, 1108, 1132, 1156, 1168, 1200, 1264, 1296, 1324, 1344, 1360, 1412, 1468, 1488, 1584, 1604, 1616, 1724

Offset: 1

Gus Wiseman, May 17 2022

nonn

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 prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    20: {1,1,3}
    36: {1,1,2,2}
    44: {1,1,5}
    56: {1,1,1,4}
    68: {1,1,7}
   100: {1,1,3,3}
   124: {1,1,11}
   164: {1,1,13}
   184: {1,1,1,9}
   208: {1,1,1,1,6}
   236: {1,1,17}
   240: {1,1,1,1,2,3}

Product of prime indices is A003963, counted by A339095.
The LHS (product of exponents) is A005361, counted by A266477.
The RHS (product of shadows) is A353394, first appearances A353397.
A related comparison is A353395, counted by A353396.
The partitions are counted by A353398.
Taking indices instead of exponents on the LHS gives A353503.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393
- recursive version counted by A353426
Cf. A000720, A003586, A005117, A143773, A182850, A316428, A320325, A324850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Times@@red/@primeMS[#]==Times@@Last/@FactorInteger[#]&]

A005361(a(n)) = A353394(a(n)).

A353931 Least run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 2, 6, 1, 2, 4, 7, 1, 8, 2, 2, 1, 9, 2, 6, 1, 6, 2, 10, 1, 11, 5, 2, 1, 3, 2, 12, 1, 2, 3, 13, 1, 14, 2, 3, 1, 15, 2, 8, 1, 2, 2, 16, 1, 3, 3, 2, 1, 17, 2, 18, 1, 4, 6, 3, 1, 19, 2, 2, 1, 20, 3, 21, 1, 2, 2, 4, 1, 22, 3, 8, 1

Offset: 1

Gus Wiseman, Jun 07 2022

nonn

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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.

Positions of first appearances are A008578.
For run-lengths instead of run-sums we have A051904, greatest A051903.
For run-sums and binary expansion we have A144790, greatest A038374.
For run-lengths and binary expansion we have A175597, greatest A043276.
Distinct run-sums are counted by A353835, weak A353861.
The greatest run-sum is given by A353862.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 3, 6, 2, 4, 1, 7, 3, 8, 4, 4, 2, 9, 3, 3, 2, 2, 5, 10, 3, 11, 1, 4, 2, 6, 2, 12, 2, 4, 4, 13, 3, 14, 6, 5, 2, 15, 3, 4, 4, 4, 7, 16, 3, 6, 5, 4, 2, 17, 5, 18, 2, 6, 1, 6, 3, 19, 8, 4, 3, 20, 3, 21, 2, 5, 9, 8, 3, 22, 4, 2, 2

Offset: 1

Gus Wiseman, Nov 28 2023

nonn

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}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

			The multiset multiplicity kernel of {1,2,2,3} is {1,1,2}, so a(90) = 4.

Positions of 1's are A000079 without 1.
Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
The triangle A367579 has these as row sums, ranks A367580.
The triangle for this rank statistic is A367582.
For maximum instead of sum we have A367583, opposite A367587.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
Cf. A000720, A005117, A051904, A055396, A061395, A071625, A072774, A130091, A175781, A367584, A367585.

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

a(n^k) = a(n) for all positive integers n and k.
a(n) = A056239(A367580(n)).
If n is squarefree, a(n) = A055396(n)*A001222(n).

A325278 Smallest number with adjusted frequency depth n.

Original entry on oeis.org

1, 2, 4, 6, 12, 60, 2520, 1286485200, 35933692027611398678865941374040400000

Offset: 0

Gus Wiseman, Apr 17 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 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.

Differs from A182857 in having 2 instead of 3.

A subsequence of A325238.
Cf. A011784, A056239, A112798, A118914, A181819, A181821, A182857, A225485, A323023, A325258, A325272, A325277, A325278, A325280.
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).

nn=10000;
fd[n_]:=Switch[n,1,0,?PrimeQ,1,,1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
fds=fd/@Range[nn];
Sort[Table[Position[fds,x][[1,1]],{x,Union[fds]}]]

A336422 Number of ways to choose a divisor of a divisor of n, both having distinct prime exponents.

Original entry on oeis.org

1, 3, 3, 6, 3, 5, 3, 10, 6, 5, 3, 13, 3, 5, 5, 15, 3, 13, 3, 13, 5, 5, 3, 24, 6, 5, 10, 13, 3, 7, 3, 21, 5, 5, 5, 21, 3, 5, 5, 24, 3, 7, 3, 13, 13, 5, 3, 38, 6, 13, 5, 13, 3, 24, 5, 24, 5, 5, 3, 20, 3, 5, 13, 28, 5, 7, 3, 13, 5, 7, 3, 42, 3, 5, 13, 13, 5, 7, 3

Offset: 1

Gus Wiseman, Jul 26 2020

nonn

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

			The a(n) ways for n = 1, 2, 4, 6, 8, 12, 30, 210:
  1/1/1  2/1/1  4/1/1  6/1/1  8/1/1  12/1/1    30/1/1  210/1/1
         2/2/1  4/2/1  6/2/1  8/2/1  12/2/1    30/2/1  210/2/1
         2/2/2  4/2/2  6/2/2  8/2/2  12/2/2    30/2/2  210/2/2
                4/4/1  6/3/1  8/4/1  12/3/1    30/3/1  210/3/1
                4/4/2  6/3/3  8/4/2  12/3/3    30/3/3  210/3/3
                4/4/4         8/4/4  12/4/1    30/5/1  210/5/1
                              8/8/1  12/4/2    30/5/5  210/5/5
                              8/8/2  12/4/4            210/7/1
                              8/8/4  12/12/1           210/7/7
                              8/8/8  12/12/2
                                     12/12/3
                                     12/12/4
                                     12/12/12

A336421 is the case of superprimorials.
A007425 counts divisors of divisors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor with distinct prime exponents.
A336500 counts divisors with quotient also having distinct prime exponents.
A336568 = not a product of two numbers with distinct prime exponents.
Cf. A000005, A001055, A005117, A032741, A071625, A118914, A124010, A336419, A336420.

strdivs[n_]:=Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&];
Table[Sum[Length[strdivs[d]],{d,strdivs[n]}],{n,30}]

cp's OEIS Frontend

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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A064839 List the natural numbers starting a new row only with each new least prime signature (A025487). a(n) is the column position associated with n.

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A329140 Numbers whose prime signature is a periodic word.

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A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

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A353394 Product of prime shadows of prime indices of n (with multiplicity).

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A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

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A353931 Least run-sum of the prime indices of n.

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A367581 Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.

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