A130092 -id:A130092 - cp's OEIS frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

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, 112, 113, 116

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

This sequence does not contain any number of the form 36n-6 or 36n+6, as such numbers are divisible by 6 but not by 4 or 9. Consequently, this sequence does not contain 24 consecutive integers. The quest for the greatest number of consecutive integers in this sequence has ties to the ABC conjecture (see the MathOverflow link). - Danny Rorabaugh, Sep 23 2015
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities. The enumeration of these partitions by sum is given by A098859. - Gus Wiseman, May 04 2019
Aktaş and Ram Murty (2017) called these terms "special numbers" ("for lack of a better word"). They prove that the number of terms below x is ~ c*x/log(x), where c > 1 is a constant. - Amiram Eldar, Feb 25 2021
Sequence A005940(1+A328592(n)), n >= 1, sorted into ascending order. - Antti Karttunen, Apr 03 2022

			From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
   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}
  25: {3,3}
  27: {2,2,2}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
MathOverflow, Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, arXiv:1902.09224 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Factorization

Complement of A130092. A351564 is the characteristic function.
Subsequence of A351294.
Cf. A000961, A006939, A181818, A304686, A319161, A342028, A342029, A342030, A342031, A342032 (subsequences).
Cf. A005940, A048767, A048768, A056239, A098859, A112798, A118914, A181796, A217605, A325326, A325337, A325368, A327498, A327523, A328592, A336423, A336424, A336569, A336570, A336571, A343012, A343013.

filter:= proc(t) local f;
f:= map2(op,2,ifactors(t)[2]);
nops(f) = nops(convert(f,set));
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 30 2015

t[n_] := FactorInteger[n][[All, 2]]; Select[Range[400],  Union[t[#]] == Sort[t[#]] &]  (* Clark Kimberling, Mar 12 2015 *)

isok(n) = {nbf = omega(n); f = factor(n); for (i = 1, nbf, for (j = i+1, nbf, if (f[i, 2] == f[j, 2], return (0)););); return (1);} \\ Michel Marcus, Aug 18 2013

isA130091(n) = issquarefree(factorback(apply(e->prime(e), (factor(n)[, 2])))); \\ Antti Karttunen, Apr 03 2022

a(n) < A130092(n) for n<=150, a(n) > A130092(n) for n>150.

A239455 Number of Look-and-Say partitions of n; see Comments.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 28, 33, 45, 55, 65, 83, 105, 121, 155, 180, 217, 259, 318, 362, 445, 512, 614, 707, 850, 958, 1155, 1309, 1543, 1754, 2079, 2327, 2740, 3085, 3592, 4042, 4699, 5253, 6093, 6815, 7839, 8751, 10069, 11208, 12832, 14266, 16270

Offset: 0

Clark Kimberling and Peter J. C. Moses, Mar 19 2014

nonn

Suppose that p = x(1) >= x(2) >= ... >= x(k) is a partition of n. Let y(1) > y(2) > ... > y(h) be the distinct parts of p, and let m(i) be the multiplicity of y(i) for 1 <= i <= h. Then we can "look" at p as "m(1) y(1)'s and m(2) y(2)'s and ... m(h) y(h)'s". Reversing the m's and y's, we can then "say" the Look-and-Say partition of p, denoted by LS(p). The name "Look-and-Say" follows the example of Look-and-Say integer sequences (e.g., A005150). As p ranges through the partitions of n, LS(p) ranges through all the Look-and-Say partitions of n. The number of these is A239455(n).
The Look-and-Say array is distinct from the Wilf array, described at A098859; for example, the number of Look-and-Say partitions of 9 is A239455(9) = 16, whereas the number of Wilf partitions of 9 is A098859(9) = 15. The Look-and-Say partition of 9 which is not a Wilf partition of 9 is [2,2,2,1,1,1].
Conjecture: a partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are all distinct, so y is counted under a(9). - Gus Wiseman, Aug 11 2025
Also the number of integer partitions y of n such that there is a pairwise disjoint way to choose a strict integer partition of each multiplicity (or run-length) of y. - Gus Wiseman, Aug 11 2025

			The 11 partitions of 6 generate 7 Look-and-Say partitions as follows:
6 -> 111111
51 -> 111111
42 -> 111111
411 -> 21111
33 -> 222
321 -> 111111
3111 -> 3111
222 -> 33
2211 -> 222
21111 -> 411
111111 -> 6,
so that a(6) counts these 7 partitions: 111111, 21111, 222, 3111, 33, 411, 6.

These include all Wilf partitions, counted by A098859, ranked by A130091.
These partitions are listed by A239454 in graded reverse-lex order.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351204, complement A351203.
The complement is counted by A351293, apparently ranked by A351295, conjugate A381433.
These partitions appear to be ranked by A351294, conjugate A381432.
The non-Wilf case is counted by A351592.
For normal multisets we appear to have A386580, complement A386581.
A000110 counts set partitions, ordered A000670.
A000569 = graphical partitions, complement A339617.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A181819 = Heinz number of the prime signature of n (prime shadow).
A279790 counts disjoint families on strongly normal multisets.
A329738 = compositions with all equal run-lengths.
A386583 counts separable partitions, sums A325534, ranks A335433.
A386584 counts inseparable partitions, sums A325535, ranks A335448.
A386585 counts separable type partitions, sums A336106, ranks A335127.
A386586 counts inseparable type partitions, sums A386638 or A025065, ranks A335126.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018, ranked by A044813.
- A329739 = compositions, for runs A351013, ranked by A351596.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A297770, A304660, A305563, A329746, A351201, A351202, A351291.

LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]]; TableForm[t = Map[LS[#] &, Range[10]]](*A239454,array*)
Flatten[t](*A239454,sequence*)
Map[Length[LS[#]] &, Range[25]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]>0&]],{n,0,10}] (* Gus Wiseman, Aug 11 2025 *)

A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

Original entry on oeis.org

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, Feb 15 2022

nonn

First differs from A130091 (Wilf partitions) in having 216.
See A239455 for the definition of Look-and-Say partitions.
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.

			The terms together with their prime indices begin:
      1: ()            20: (3,1,1)         47: (15)
      2: (1)           23: (9)             48: (2,1,1,1,1)
      3: (2)           24: (2,1,1,1)       49: (4,4)
      4: (1,1)         25: (3,3)           50: (3,3,1)
      5: (3)           27: (2,2,2)         52: (6,1,1)
      7: (4)           28: (4,1,1)         53: (16)
      8: (1,1,1)       29: (10)            54: (2,2,2,1)
      9: (2,2)         31: (11)            56: (4,1,1,1)
     11: (5)           32: (1,1,1,1,1)     59: (17)
     12: (2,1,1)       37: (12)            61: (18)
     13: (6)           40: (3,1,1,1)       63: (4,2,2)
     16: (1,1,1,1)     41: (13)            64: (1,1,1,1,1,1)
     17: (7)           43: (14)            67: (19)
     18: (2,2,1)       44: (5,1,1)         68: (7,1,1)
     19: (8)           45: (3,2,2)         71: (20)
For example, the prime indices of 216 are {1,1,1,2,2,2}, and there are four permutations with distinct run-lengths: (1,1,2,2,2,1), (1,2,2,2,1,1), (2,1,1,1,2,2), (2,2,1,1,1,2); so 216 is in the sequence. It is the Heinz number of the Look-and-Say partition of (3,3,2,1).

The Wilf case (distinct multiplicities) is A130091, counted by A098859.
The complement of the Wilf case is A130092, counted by A336866.
These partitions appear to be counted by A239455.
A variant for runs is A351201, counted by A351203 (complement A351204).
The complement is A351295, counted by A351293.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of run-lengths in binary expansion, for all runs A297770.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000961, A001221, A001222, A047966, A175413, A182857, A304660, A320924, A328592, A329747, A351202, A351290, A351592.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]!={}&]

Name edited by Gus Wiseman, Aug 13 2025

A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 28, 44, 56, 80, 111, 148, 192, 264, 335, 447, 575, 743, 937, 1213, 1513, 1924, 2396, 3011, 3715, 4646, 5687, 7040, 8600, 10556, 12804, 15650, 18897, 22930, 27593, 33296, 39884, 47921, 57168, 68360, 81295, 96807, 114685

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

First differs from A336866 (non-Wilf partitions) at a(9) = 14, A336866(9) = 15, the difference being the partition (2,2,2,1,1,1).

See A239455 for the definition of Look-and-Say partitions.

			The a(3) = 1 through a(9) = 14 partitions:
  (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)
                                             (321111)

The complement is counted by A239455, ranked by A351294.
These are all non-Wilf partitions (counted by A336866, ranked by A130092).
A variant for runs is A351203, complement A351204, ranked by A351201.
These partitions appear to be ranked by A351295.
Non-Wilf partitions in the complement are counted by A351592.
A000569 = graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A098859 = Wilf partitions (distinct multiplicities), ranked by A130091.
A181819 = Heinz number of the prime signature of n (prime shadow).
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A238130, A297770, A304660, A305563, A329740, A329746, A351202, A351291.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]==0&]],{n,0,15}] (* Gus Wiseman, Aug 13 2025 *)

a(n) = A000041(n) - A239455(n).

Edited by Gus Wiseman, Aug 12 2025

A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

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.

			The terms together with their prime indices begin:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]

Name edited by Gus Wiseman, Aug 13 2025

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

Original entry on oeis.org

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

A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

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, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A130092 in lacking 180.
First differs from A351295 in lacking 180 and having 216.
First differs from A362605 in having 60.

			The prime factorization of 1800 is {2,2,2,3,3,5,5}, and the parts of least multiplicity are {3,5}, so 1800 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}

The complement is A359178, counted by A362610.
For mode we have A362605, counted by A362607.
Partitions of this type are counted by A362609.
These are the positions of terms > 1 in A362613.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes.
A362615 counts partitions by number of co-modes.
Cf. A215366, A327473, A327476, A353864, A356862, A359908, A362611.

Select[Range[100],Count[Last/@FactorInteger[#],Min@@Last/@FactorInteger[#]]>1&]

A383512 Heinz numbers of conjugate Wilf partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
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: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

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

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
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.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

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

A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

Original entry on oeis.org

13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

The n-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 n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
  13:     1101  (1,2,1)
  22:    10110  (2,1,2)
  25:    11001  (1,3,1)
  45:   101101  (2,1,2,1)
  46:   101110  (2,1,1,2)
  49:   110001  (1,4,1)
  53:   110101  (1,2,2,1)
  54:   110110  (1,2,1,2)
  59:   111011  (1,1,2,1,1)
  76:  1001100  (3,1,3)
  77:  1001101  (3,1,2,1)
  82:  1010010  (2,3,2)
  89:  1011001  (2,1,3,1)
  91:  1011011  (2,1,2,1,1)
  93:  1011101  (2,1,1,2,1)
  94:  1011110  (2,1,1,1,2)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers of partitions is A130092, complement A130091.
Normal multisets with a permutation of this type appear to be A283353.
Partitions w/o permutations of this type are A351204, complement A351203.
The version using binary expansions is A351205, complement A175413.
The complement is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A345167 ranks alternating compositions, counted by A025047.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]

cp's OEIS Frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

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A239455 Number of Look-and-Say partitions of n; see Comments.

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A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

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A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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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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A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

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