A336866 -id:A336866 - cp's OEIS frontend

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

325, 845, 931, 1625, 2527, 3509, 6253, 6517, 8125, 9251

Offset: 1

Gus Wiseman, May 18 2025

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 Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
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.
A integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its positive 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
   325: {3,3,6}
   845: {3,6,6}
   931: {4,4,8}
  1625: {3,3,3,6}
  2527: {4,8,8}
  3509: {5,5,10}
  6253: {6,6,12}
  6517: {4,4,4,8}
  8125: {3,3,3,3,6}
  9251: {5,10,10}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383511.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
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 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).
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. A047966, A051903, A212166, A238745, A325368, A351592, 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[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]

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.

A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 20 2025

Also differences of distinct 0-prepended prime indices of n.

			The prime indices of 140 are {1,1,3,4}, zero prepended {0,1,1,3,4}, differences (1,0,2,1), positive (1,2,1).
Rows begin:
    1: ()        16: (1)        31: (11)
    2: (1)       17: (7)        32: (1)
    3: (2)       18: (1,1)      33: (2,3)
    4: (1)       19: (8)        34: (1,6)
    5: (3)       20: (1,2)      35: (3,1)
    6: (1,1)     21: (2,2)      36: (1,1)
    7: (4)       22: (1,4)      37: (12)
    8: (1)       23: (9)        38: (1,7)
    9: (2)       24: (1,1)      39: (2,4)
   10: (1,2)     25: (3)        40: (1,2)
   11: (5)       26: (1,5)      41: (13)
   12: (1,1)     27: (2)        42: (1,1,2)
   13: (6)       28: (1,3)      43: (14)
   14: (1,3)     29: (10)       44: (1,4)
   15: (2,1)     30: (1,1,1)    45: (2,1)

Row-lengths are A001221, sums A061395.
Rows start with A055396, end with A241919.
For multiplicities instead of differences we have A124010 (prime signature).
Including difference 0 gives A287352, without prepending A355536.
Positions of first appearances of rows are A358137.
Positions of strict rows are A383512, counted by A098859.
Positions of non-strict rows are A383513, counted by A336866.
Heinz numbers of rows are A383535.
Restricting to rows of squarefree index gives A384008.
Without prepending we get A384009.
A000040 lists the primes, differences A001223.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A005117, A122111, A130091, A325325, A325349, A325366, A325368, A381431, A383506.

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

a(A005117(n)) = A384008(n).

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}]

A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 23, 25, 43, 63, 345, 365, 665, 949, 1513, 8175, 9003, 15929, 23399, 36949, 51043, 293715, 314697, 570353, 826817, 1318201, 1810393, 2766099, 14180139, 15600413, 27707879, 40501321, 63981955, 88599903, 134362569, 181491125, 923029217

Offset: 0

Gus Wiseman, Jun 19 2025

nonn

			The permutation (1,2,6,7,8,9,3,4,5) has maximal runs ((1,2),(6,7,8,9),(3,4,5)), with lengths (2,4,3), so is counted under a(9).
The a(0) = 1 through a(7) = 25 permutations:
  ()  (1)  (12)  (123)  (1234)  (12345)  (123456)  (1234567)
                 (231)  (2341)  (23451)  (123564)  (1234675)
                 (312)  (4123)  (34512)  (123645)  (1234756)
                                (45123)  (124563)  (1245673)
                                (51234)  (126345)  (1273456)
                                         (145623)  (1456723)
                                         (156234)  (1672345)
                                         (231456)  (2314567)
                                         (234156)  (2345167)
                                         (234561)  (2345671)
                                         (312456)  (3124567)
                                         (345126)  (3456127)
                                         (345612)  (3456712)
                                         (412356)  (4567123)
                                         (451236)  (4567231)
                                         (456231)  (4567312)
                                         (456312)  (5123467)
                                         (561234)  (5612347)
                                         (562341)  (5671234)
                                         (564123)  (6712345)
                                         (612345)  (6723451)
                                         (634512)  (6751234)
                                         (645123)  (7123456)
                                                   (7345612)
                                                   (7561234)

Christian Sievers, Table of n, a(n) for n = 0..5000

Counting by number of maximal anti-runs gives A010027, for runs A123513.
For subsets instead of permutations we have A384175, complement A384176.
For partitions we have A384884 (anti-runs A384885), strict A384178 (anti-runs A384880).
For equal instead of distinct lengths we have A384892.
For anti-runs instead of runs we have A384907.
A000041 counts integer partitions, strict A000009.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A287170, A325324, A325325, A328592, A329739, A351202, A384177, A384886.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

lista(n)=my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(n+1))/(1-x))+O(x*x^d))); Vec(1+sum(i=1,d,i!*b(i)*polcoef(p,i))) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(k-1) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574). - Christian Sievers, Jun 22 2025

a(11) and beyond from Christian Sievers, Jun 22 2025

A384892 Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 2, 4, 13, 54, 314, 2120, 16700, 148333, 1468512, 16019532, 190899736, 2467007774, 34361896102, 513137616840, 8178130784179, 138547156531410, 2486151753462260, 47106033220679060, 939765362754015750, 19690321886243848784, 432292066866187743954

Offset: 0

Gus Wiseman, Jun 19 2025

nonn

			The permutation (1,2,5,6,3,4,7,8) has maximal runs ((1,2),(5,6),(3,4),(7,8)), with lengths (2,2,2,2), so is counted under a(8).
The a(0) = 1 through a(4) = 13 permutations:
  ()  (1)  (12)  (123)  (1234)
           (21)  (132)  (1324)
                 (213)  (1432)
                 (321)  (2143)
                        (2413)
                        (2431)
                        (3142)
                        (3214)
                        (3241)
                        (3412)
                        (4132)
                        (4213)
                        (4321)

Christian Sievers, Table of n, a(n) for n = 0..450

For subsets instead of permutations we have A243815, for anti-runs A384889.
For strict partitions and distinct lengths we have A384178, anti-runs A384880.
For integer partitions and distinct lengths we have A384884, anti-runs A384885.
For distinct lengths we have A384891, for anti-runs A384907.
For partitions we have A384904, strict A384886.
A010027 counts permutations by maximal anti-runs, for runs A123513.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A047993, A242882, A325325, A329739, A351202, A384175, A384176, A384177.

Table[Length[Select[Permutations[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

a(n)=if(n,sumdiv(n,d,sum(i=0,d-1,(-1)^i*(d-i)!*binomial(d-1,i))),1) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{d|n} A000255(d-1). - Christian Sievers, Jun 22 2025

a(11) and beyond from Christian Sievers, Jun 22 2025

A384904 Number of integer partitions of n with all equal lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 9, 9, 14, 17, 23, 25, 40, 41, 59, 68, 92, 99, 140, 151, 204, 229, 296, 328, 433, 476, 606, 685, 858, 955, 1203, 1336, 1654, 1858, 2266, 2537, 3102, 3453, 4169, 4680, 5611, 6262, 7495, 8358, 9927, 11105, 13096, 14613, 17227, 19179, 22459

Offset: 0

Gus Wiseman, Jun 20 2025

nonn

			The partition (6,5,5,4,2,1) has maximal runs ((6,5),(5,4),(2,1)), with lengths (2,2,2), so is counted under a(23).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (311)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (511)      (422)
                                     (411)     (4111)     (611)
                                     (3111)    (31111)    (2222)
                                     (111111)  (1111111)  (3221)
                                                          (3311)
                                                          (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..100

For subsets instead of strict partitions we have A243815, distinct lengths A384175.
For distinct instead of equal lengths we have A384882, counting gaps of 0 A384884.
The strict case is A384886.
Counting gaps of 0 gives A384887.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
Cf. A000217, A008284, A047966, A089259, A325325, A382857, A383013, A383708, A384178, A384880.

Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]

A_q(N) = {Vec(1+sum(k=1,floor(-1/2+sqrt(2+2*N)), sum(i=1,(N/(k*(k+1)/2))+1, q^((k*i*(2+i*(k-1)))/2)/(1-q^(k*i))*prod(j=1,i-1, 1 + q^(2*k*j)/(1 - q^(k*j))))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 20 2025

G.f.: 1 + Sum_{i,k>0} q^((i*k*(2 + i*(k-1)))/2) * Product_{j=1..i-1} ( 1 + q^(2*k*j)/(1 - q^(k*j)) ) / (1 - q^(i*k)). - John Tyler Rascoe, Aug 20 2025

A336425 Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.

Original entry on oeis.org

1, 1, 3, 5, 24, 38, 132, 195, 570, 1588, 4193, 6086, 14561, 19232, 37142, 106479, 207291, 266871, 549726, 674330, 1465399, 3086598, 5939574, 7182133, 12324512, 28968994, 46819193, 82873443, 165205159, 196666406, 350397910, 406894074, 593725529, 1229814478, 1853300600, 4024414209, 6049714096, 6968090487, 9700557121, 16810076542, 26339337285

Offset: 0

Gus Wiseman, Aug 06 2020

nonn

			The a(4) = 24 divisors of divisors:
  1/1  2/1  3/1  4/1  8/1  12/1   24/1
       2/2  3/3  4/2  8/2  12/2   24/2
                 4/4  8/4  12/3   24/3
                      8/8  12/4   24/4
                           12/12  24/8
                                  24/12
                                  24/24

Max Alekseyev, Table of n, a(n) for n = 0..85

A336422 is the non-factorial generalization.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327526 gives the maximum divisor of n with equal prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
A336414 counts divisors of n! with distinct prime exponents.
A336415 counts divisors of n! with equal prime exponents.
A336423 counts chains in A130091, with maximal version A336569.
Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568, A336570, A336571, A336865, A336866, A336869.
Factorial numbers: A000142, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327499, A336416, A336418, A336617.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[n!],d_?strsigQ:>Count[Divisors[d],e_?strsigQ]]],{n,0,20}]

Terms a(21) onward from Max Alekseyev, Nov 07 2024

cp's OEIS Frontend

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

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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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A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

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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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A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).

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A384892 Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).

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