A130092 -id:A130092 - cp's OEIS frontend

A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset: 1

Gus Wiseman, Apr 18 2025

nonn

First differs from A359178 (complement A362606) in having 1, 240 and lacking 180.
First differs from A130091 (complement A130092) in having 240 and lacking 360.
First differs from A351294 (complement A351295) in having 240 and lacking 216.
Includes all primes A000040 and prime powers A000961.
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, sum A056239.

			The prime indices of 144 are {1,1,1,1,2,2}, with just one permutation with all equal run-lengths (1,1,2,2,1,1), so 144 is in the sequence.
The prime indices of 240 are {1,1,1,1,2,3}, which have no permutation with all equal run-lengths, so 240 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}

These are positions of zeros and ones in A382857, just zeros A382879, just ones A383112.
The complement for run-sums instead of lengths is A383015, counted by A383097.
The complement is A383089, counted by A383090.
Partitions of this type are counted by A383092, just zero A382915, just one A383094.
For run-sums instead of lengths we have A383099 \/ A383100, counted by A383095 + A383096.
A047966 counts partitions with equal run-lengths, compositions A329738.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A000961, A001221, A001222, A048767, A351294, A353744, A353833, A381435, A382771, A382877, A383113.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]<=1&]

Equals A382879 \/ A383112, counted by A382915 + A383094.

A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 15 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Partitions of this type are counted by A383507.
Negating both sides gives A383531, counted by A383530.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
paug[y_]:=-DeleteCases[Differences[Append[y,0]],0];
Select[Range[100], UnsameQ@@Last/@FactorInteger[#] && UnsameQ@@paug[Reverse[prix[#]]]&]

Equals A130091 /\ A383512.

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713

Offset: 0

Gus Wiseman, May 14 2025

nonn

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

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

These partitions have Heinz numbers A130091 /\ A383512.

A383514 Heinz numbers of non Wilf section-sum partitions.

Original entry on oeis.org

10, 14, 15, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190

Offset: 1

Gus Wiseman, May 18 2025

nonn

First differs from A384007 in having 1000.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The terms together with their prime indices begin:
    10: {1,3}    57: {2,8}      94: {1,15}
    14: {1,4}    58: {1,10}     95: {3,8}
    15: {2,3}    62: {1,11}    100: {1,1,3,3}
    22: {1,5}    65: {3,6}     106: {1,16}
    26: {1,6}    69: {2,9}     111: {2,12}
    33: {2,5}    74: {1,12}    115: {3,9}
    34: {1,7}    77: {4,5}     118: {1,17}
    35: {3,4}    82: {1,13}    119: {4,7}
    38: {1,8}    85: {3,7}     122: {1,18}
    39: {2,6}    86: {1,14}    123: {2,13}
    46: {1,9}    87: {2,10}    129: {2,14}
    51: {2,7}    91: {4,6}     130: {1,3,6}
    55: {3,5}    93: {2,11}    133: {4,8}

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A351592 (A384006).
These partitions are counted by A383506.
The Look-and-Say case is A383511 (A383518).
For Wilf instead of non Wilf we have A383519 (A383520).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 21, 27, 30, 33, 41, 50, 57, 68, 79, 89, 112, 126, 144, 172, 198, 220, 257, 298, 327, 383, 423, 477, 533, 621, 650, 760, 816, 920, 1013

Offset: 0

Gus Wiseman, May 19 2025

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A098859 (A130091), conjugate (A383512).
For non Wilf instead of Wilf we have A383506 (A383514).
These partitions are ranked by (A383520).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
Cf. A047966, A320348, A325324, A325325, A351592, A353837, A381431, A383530, A383709, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],disjointFamilies[conj[#]]!={}&&UnsameQ@@Length/@Split[#]&]],{n,0,15}]

A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 20, 23, 25, 28, 29, 31, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 121, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164

Offset: 1

Gus Wiseman, May 15 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Integer partitions with distinct multiplicities are called Wilf partitions.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}
   47: {15}
   49: {4,4}
   50: {1,3,3}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

For just distinct multiplicities we have A130091 (conjugate A383512), counted by A098859.
For just distinct 0-appended differences we have A325367, counted by A325324.
These partitions are counted by A383709.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
Cf. A000720, A005117, A047966, A238745, A320348, A325325, A325349, A325355, A325366, A325368, A325388, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Length/@Split[prix[#]] && UnsameQ@@Differences[Append[Reverse[prix[#]],0]]&]

Equals A130091 /\ A325367.

A136567 a(n) is the number of exponents occurring only once each in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Jan 07 2008

nonn

Records are in A006939: 1, 2, 12, 360, 75600, ..., . - Robert G. Wilson v, Jan 20 2008

			4200 = 2^3 * 3^1 * 5^2 * 7^1. The exponents of the prime factorization are therefore 3,1,2,1. The exponents occurring exactly once are 2 and 3. So a(4200) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A056169, A071625, A133924, A136566, A181819.

For a(n)=0 see A130092 plus the term 1; for a(n)=1 see A000961.

f[n_] := Block[{fi = Sort[Last /@ FactorInteger@n]}, Count[ Count[fi, # ] & /@ Union@fi, 1]]; f[1] = 0; Array[f, 105] (* Robert G. Wilson v, Jan 20 2008 *)
Table[Boole[n != 1] Count[Split@ Sort[FactorInteger[n][[All, -1]]], ?(Length@ # == 1 &)], {n, 105}] (* _Michael De Vlieger, Jul 24 2017 *)

A136567(n) = { my(exps=(factor(n)[, 2]), m=prod(i=1, length(exps), prime(exps[i])), f=factor(m)[, 2]); sum(i=1, #f, f[i]==1); }; \\ Antti Karttunen, Jul 24 2017

(define (A136567 n) (A056169 (A181819 n))) ;; Antti Karttunen, Jul 24 2017

a(n) = A056169(A181819(n)). - Antti Karttunen, Jul 24 2017

More terms from Robert G. Wilson v, Jan 20 2008

A336867 Numbers k such that k! does not have distinct prime multiplicities.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Aug 07 2020

nonn

The complement appears to be {0, 1, 2, 4, 6, 10}.
A number has distinct prime multiplicities iff its prime signature is strict.
From Chai Wah Wu, Aug 11 2020: (Start)
Theorem: the sequence consists of all nonnegative integers except 0, 1, 2, 4, 6, 10.
Proof: The cases k <= 31 follow from inspection. We show the case where k > 31.
Note that if p < q are successive primes, then for q <= m < 2p, the multiplicities of p and q in m! are both 1, i.e., m is a term.
Assume that p >= 29. Nagura showed that for all k >= 25 there exists a prime x such that  k < x < 1.2k. This implies that q < 1.2p and thus 2p > 1.666q, i.e. for q <= m < 1.666q, m is a term.
Again by Nagura's theorem, there exists a prime r < 1.2q. Thus intervals of the form [q, 1.666q] for q prime span all integers > 31 and the result is proved. QED
(End)

			The sequence of indexed factorials a(n)! together with their prime signatures begins:
                    6: (1,1)
                  120: (3,1,1)
                 5040: (4,2,1,1)
                40320: (7,2,1,1)
               362880: (7,4,1,1)
             39916800: (8,4,2,1,1)
            479001600: (10,5,2,1,1)
           6227020800: (10,5,2,1,1,1)
          87178291200: (11,5,2,2,1,1)
        1307674368000: (11,6,3,2,1,1)
       20922789888000: (15,6,3,2,1,1)
      355687428096000: (15,6,3,2,1,1,1)
     6402373705728000: (16,8,3,2,1,1,1)
   121645100408832000: (16,8,3,2,1,1,1,1)
  2432902008176640000: (18,8,4,2,1,1,1,1)

J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952) 177-181.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A130092 is the generalization to non-factorials.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336866 counts partitions without distinct multiplicities.
Cf. A098859, A118914, A124010, A336423, A336424, A336500, A336568, A336571.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416, A336869.

Select[Range[0,100],!UnsameQ@@Last/@FactorInteger[#!]&]

From Chai Wah Wu, Aug 11 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 7.
G.f.: x*(-x^6 + x^5 - x^3 - x + 3)/(x - 1)^2. (End)

A336865 Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 06 2020

Row lengths are A073093(n) = A001222(n) + 1.

			The triangle begins as follows. The n-th row is shown to the right of "n:".
     1: (1)          16: (1,1,1,1,1)    31: (1,1)
     2: (1,1)        17: (1,1)          32: (1,1,1,1,1,1)
     3: (1,1)        18: (1,2,1,1)      33: (1,2,0)
     4: (1,1,1)      19: (1,1)          34: (1,2,0)
     5: (1,1)        20: (1,2,1,1)      35: (1,2,0)
     6: (1,2,0)      21: (1,2,0)        36: (1,2,2,2,0)
     7: (1,1)        22: (1,2,0)        37: (1,1)
     8: (1,1,1,1)    23: (1,1)          38: (1,2,0)
     9: (1,1,1)      24: (1,2,1,2,1)    39: (1,2,0)
    10: (1,2,0)      25: (1,1,1)        40: (1,2,1,2,1)
    11: (1,1)        26: (1,2,0)        41: (1,1)
    12: (1,2,1,1)    27: (1,1,1,1)      42: (1,3,0,0)
    13: (1,1)        28: (1,2,1,1)      43: (1,1)
    14: (1,2,0)      29: (1,1)          44: (1,2,1,1)
    15: (1,2,0)      30: (1,3,0,0)      45: (1,2,1,1)
Row n = 72 counts the following divisors:
  1  2  4   8  24  72
     3  9  12
           18
Row n = 1200 counts the following divisors:
  1  2   4   8  16   48  400  1200
     3  25  12  24   80  600
     5      20  40  200
            50
            75

A073093 gives row lengths.
A130092 gives positions of rows ending with 0.
A146291 is the version not requiring distinct prime multiplicities.
A181796 gives row sums.
A336499 is the restriction to factorial numbers.
A001222 counts prime factors, counting multiplicity.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336423 counts chains using A130091.
Cf. A000005, A001221, A098859, A118914, A124010, A336422, A336500, A336571.

Table[Length[Select[Divisors[n],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,20},{k,0,PrimeOmega[n]}]

A353693 a(n) is the least multiplier k such that the exponents in the prime factorization of k*n are mutually distinct (A130091).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 3, 2, 5, 2, 1, 2, 3, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 2, 1, 6, 1, 2, 1, 1, 5, 12, 1, 1, 3, 20, 1, 1, 1, 2, 1, 1, 7, 12, 1, 1, 1, 2, 1, 6, 5, 2

Offset: 1

Amiram Eldar, May 04 2022

nonn

First differs from A327499 at n = 30.
If n = Product_{i=1..k} p_i is squarefree (A005117), and p_1 < p_2 < ... < p_k are its k ordered prime divisors, then a(n) = Product_{i} p_i^(k-i).
If n is powerful (A001694) then a(n) = a(n/rad(n)), where rad(n) is the squarefree kernel of n (A007947). In general, if k = A051904(n) is the minimal exponent in the prime factorization of n, then a(n) = a(n/(rad(n)^(k-1))).

			a(2) = 1 since 2 = 2^1 has only one exponent (1) in its prime factorization.
a(6) = 2 since 6 = 2*3 has two equal exponents (1) in its prime factorization, and 2*6 = 12 = 2^2*3 has two distinct exponents (1 and 2).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A005117, A007947, A130091, A130092, A327498, A327499, A353694.

a[n_] := Module[{k = 1}, While[!UnsameQ @@ FactorInteger[k*n][[;; , 2]], k++]; k]; Array[a, 100]

a(n) = my(k=1, f=factor(n)[,2]); while(#Set(f) != #f, k++; f=factor(k*n)[,2]); k; \\ Michel Marcus, May 05 2022

a(n) = 1 if and only if n is in A130091.
a(A130092(n)) > 1.
rad(a(n)) | rad(n).
a(n) = A353694(n)/n.

cp's OEIS Frontend

A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

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A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

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A383507 Number of Wilf and conjugate Wilf integer partitions of n.

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A383514 Heinz numbers of non Wilf section-sum partitions.

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A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

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A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

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A136567 a(n) is the number of exponents occurring only once each in the prime factorization of n.

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A336867 Numbers k such that k! does not have distinct prime multiplicities.

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