A091602 -id:A091602 - cp's OEIS frontend

A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 6, 0, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 13, 1, 0, 0, 0, 0, 1, 0, 16, 4, 0, 1, 0, 0, 0, 1, 0, 25, 2, 2, 0, 0, 0, 0, 0, 1, 0, 33, 6, 1, 0, 1, 0, 0, 0, 0, 1, 0, 49, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 61, 9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 90, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 156, 9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Jun 28 2014

T(0,0) = 1 by convention.
Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041.
Row sums are A000041.

			Triangle starts:
00:  1;
01:  0,   1;
02:  0,   1,  1;
03:  0,   2,  0, 1;
04:  0,   3,  1, 0, 1;
05:  0,   6,  0, 0, 0, 1;
06:  0,   7,  2, 1, 0, 0, 1;
07:  0,  13,  1, 0, 0, 0, 0, 1;
08:  0,  16,  4, 0, 1, 0, 0, 0, 1;
09:  0,  25,  2, 2, 0, 0, 0, 0, 0, 1;
10:  0,  33,  6, 1, 0, 1, 0, 0, 0, 0, 1;
11:  0,  49,  4, 2, 0, 0, 0, 0, 0, 0, 0, 1;
12:  0,  61,  9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
13:  0,  90,  6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
14:  0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
15:  0, 156,  9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
16:  0, 198, 23, 3, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
17:  0, 269, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18:  0, 334, 34, 9, 3, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
19:  0, 448, 27, 8, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20:  0, 556, 51, 7, 6, 3, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
The A000041(9) = 30 partitions of 9 with the least multiplicities of any part are:
01:  [ 1 1 1 1 1 1 1 1 1 ]   9
02:  [ 1 1 1 1 1 1 1 2 ]   1
03:  [ 1 1 1 1 1 1 3 ]   1
04:  [ 1 1 1 1 1 2 2 ]   2
05:  [ 1 1 1 1 1 4 ]   1
06:  [ 1 1 1 1 2 3 ]   1
07:  [ 1 1 1 1 5 ]   1
08:  [ 1 1 1 2 2 2 ]   3
09:  [ 1 1 1 2 4 ]   1
10:  [ 1 1 1 3 3 ]   2
11:  [ 1 1 1 6 ]   1
12:  [ 1 1 2 2 3 ]   1
13:  [ 1 1 2 5 ]   1
14:  [ 1 1 3 4 ]   1
15:  [ 1 1 7 ]   1
16:  [ 1 2 2 2 2 ]   1
17:  [ 1 2 2 4 ]   1
18:  [ 1 2 3 3 ]   1
19:  [ 1 2 6 ]   1
20:  [ 1 3 5 ]   1
21:  [ 1 4 4 ]   1
22:  [ 1 8 ]   1
23:  [ 2 2 2 3 ]   1
24:  [ 2 2 5 ]   1
25:  [ 2 3 4 ]   1
26:  [ 2 7 ]   1
27:  [ 3 3 3 ]   3
28:  [ 3 6 ]   1
29:  [ 4 5 ]   1
30:  [ 9 ]   1
Therefore row n=9 is [0, 25, 2, 2, 0, 0, 0, 0, 0, 1].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)

Cf. A183568, A242451 (the same for compositions).

Cf. A091602 (partitions by max multiplicity of any part).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
    end:
T:= (n, k)-> b(n$2, k) -`if`(n=0 and k=0, 0, b(n$2, k+1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-i*j, i-1, k], {j, Max[1, k], n/i}]]]; T[n_, k_] := b[n, n, k] - If[n == 0 && k == 0, 0, b[n, n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)

A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 6, 4, 1, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 3, 1, 1, 3, 2, 5, 1, 9, 2, 1, 1, 1, 1, 3, 6, 1, 1, 1, 2, 4, 1, 1, 10, 2, 2, 1, 11, 1, 1, 1, 1, 1, 4, 2, 7, 1, 2, 3, 1, 2, 1, 1, 12, 8, 1, 5, 2, 3, 1, 1, 1

Offset: 2

Gus Wiseman, Jul 12 2022

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.
The augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 2 1
   7: 4
   8: 1 1 1
   9: 1 2
  10: 3 1
  11: 5
  12: 2 1 1
  13: 6
  14: 4 1
  15: 2 2
  16: 1 1 1 1
For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).

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

Crossrefs found in the link are not repeated here.
Row-lengths are A001222.
Row-sums are A252464
Other similar triangles are A287352, A091602.
Constant rows have indices A307824.
The Heinz numbers of the rows are A325351.
Strict rows have indices A325366.
Row minima are A355531, non-augmented A355524, also A355525.
Row maxima are A355535, non-augmented A286470, also A355526.
The non-augmented version is A355536, also A355533.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 28 2025

Row lengths are A066328.
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.
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The prime indices of 24 are (2,1,1,1), with Look-and-Say partition (3,1,1), so row 24 is (3,1,1).
The prime indices of 36 are (2,2,1,1), with Look-and-Say partition (2,2,2), so row 36 is (2,2,2).
Triangle begins:
   1: (empty)
   2: 1
   3: 1 1
   4: 2
   5: 1 1 1
   6: 1 1 1
   7: 1 1 1 1
   8: 3
   9: 2 2
  10: 1 1 1 1
  11: 1 1 1 1 1
  12: 2 1 1
  13: 1 1 1 1 1 1
  14: 1 1 1 1 1
  15: 1 1 1 1 1
  16: 4
  17: 1 1 1 1 1 1 1
  18: 2 2 1
  19: 1 1 1 1 1 1 1 1

Heinz numbers are A048767 (union A351294, complement A351295, fixed A048768, A217605).
First part in each row is A051903, conjugate A066328.
Last part in each row is A051904, conjugate A381437 (counted by A381438).
Row sums are A056239.
Row lengths are A066328.
Partitions of this type are counted by A239455, complement A351293.
The conjugate is A381436, Heinz numbers A381431 (union A381432, complement A381433).
Rows appearing only once have Heinz numbers A381540, more than once A381541.
A000040 lists the primes.
A003963 gives product of prime indices.
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.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A071178, A091602, A116861, A130091, A181819, A212166, A238744, A380955.

Table[Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>ConstantArray[k,PrimePi[p]]]]//Reverse,{n,30}]

A213177 Number T(n,k) of parts in all partitions of n with largest multiplicity k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 3, 0, 3, 0, 3, 5, 0, 4, 0, 5, 6, 4, 0, 5, 0, 8, 9, 7, 5, 0, 6, 0, 10, 13, 13, 5, 6, 0, 7, 0, 13, 23, 14, 15, 6, 7, 0, 8, 0, 18, 30, 27, 16, 13, 7, 8, 0, 9, 0, 25, 44, 33, 30, 18, 15, 8, 9, 0, 10, 0, 30, 58, 55, 36, 34, 15, 17, 9, 10, 0, 11

Offset: 0

Alois P. Heinz, Feb 27 2013

			T(6,1) = 8: partitions of 6 with largest multiplicity 1 are [3,2,1], [4,2], [5,1], [6], with 3+2+2+1 = 8 parts.
T(6,2) = 9: [2,2,1,1], [3,3], [4,1,1].
T(6,3) = 7: [2,2,2], [3,1,1,1].
T(6,4) = 5: [2,1,1,1,1].
T(6,5) = 0.
T(6,6) = 6: [1,1,1,1,1,1].
Triangle begins:
  0;
  0,  1;
  0,  1,  2;
  0,  3,  0,  3;
  0,  3,  5,  0,  4;
  0,  5,  6,  4,  0,  5;
  0,  8,  9,  7,  5,  0,  6;
  0, 10, 13, 13,  5,  6,  0,  7;
  0, 13, 23, 14, 15,  6,  7,  0,  8;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000004, A015723, A320372, A320373, A320374, A320375, A320376, A320377, A320378, A320379, A320380.
Row sums give: A006128.
Main diagonal and first lower diagonal give: A001477, A063524.
T(2n,n) gives A320381.
Cf. A091602, A210485.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2] -b(n, n, k-1)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[, 0] = 0; T[n, k_] := b[n, n, k][[2]] - b[n, n, k-1][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = A210485(n,k) - A210485(n,k-1) for k>0, T(n,0) = 0.

A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Jul 10 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.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 4.

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

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of 0's are A025475, minimal version A013929.
Positions of 1's are 2 followed by A066312, minimal version A355527.
Triangle A238710 counts m such that A056239(m) = n and a(m) = k.
Prepending 0 to the prime indices gives A286469, minimal version A355528.
See also A286470, minimal version A355524.
The minimal version is A355525, triangle A238709.
The augmented version is A355532.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A047966, A091602, A238353, A279945, A325160, A325161, A352822, A351294, A355530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],PrimePi[n],Max@@Differences[primeMS[n]]],{n,2,100}]

A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 1, 4, 3, 3, 2, 1, 0, 1, 3, 5, 4, 4, 3, 1, 1, 0, 1, 2, 6, 4, 8, 3, 3, 2, 1, 0, 1, 3, 7, 9, 6, 7, 4, 3, 1, 1, 0, 1, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1

Offset: 0

Gus Wiseman, Nov 28 2023

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.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  1  1
  0  1  1  2  2  1
  0  1  3  3  2  1  1
  0  1  1  4  3  3  2  1
  0  1  3  5  4  4  3  1  1
  0  1  2  6  4  8  3  3  2  1
  0  1  3  7  9  6  7  4  3  1  1
  0  1  1  8  7 11  9  9  4  3  2  1
  0  1  5 10 11 13 10 11  6  5  3  1  1
  0  1  1 10 11 17 14 18 10  9  4  3  2  1
  0  1  3 12 17 19 18 22 14 12  8  4  3  1  1
  0  1  3 12 15 27 19 31 19 19 10  9  5  3  2  1
  0  1  4 15 23 27 31 33 24 26 18 12  8  4  3  1  1
  0  1  1 14 20 35 33 48 32 37 25 20 11 10  4  3  2  1
Row n = 7 counts the following partitions:
  (1111111)  (61)  (421)     (52)     (4111)  (511)  (7)
                   (2221)    (331)    (322)   (43)
                   (22111)   (31111)  (3211)
                   (211111)

Column k = 2 is A000005(n) - 1 = A032741(n).
Row sums are A000041.
The case of constant partitions is A051731, row sums A000005.
The corresponding rank statistic is A367581, row sums of A367579.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A056239, A066328, A071625, A072774, A082090, A367580.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n,0,10}, {k,0,n}]

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 3, 1, 0, 0, 3, 4, 1, 2, 0, 0, 4, 7, 2, 1, 0, 0, 0, 5, 9, 4, 1, 2, 0, 0, 0, 6, 13, 4, 4, 1, 0, 0, 0, 0, 8, 18, 6, 3, 2, 3, 0, 0, 0, 0, 10, 26, 9, 5, 2, 2, 0, 0, 0, 0, 0, 12, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset: 1

Gus Wiseman, Mar 01 2025

The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Triangle begins:
   1
   1  1
   1  0  2
   2  1  0  2
   3  1  0  0  3
   4  1  2  0  0  4
   7  2  1  0  0  0  5
   9  4  1  2  0  0  0  6
  13  4  4  1  0  0  0  0  8
  18  6  3  2  3  0  0  0  0 10
  26  9  5  2  2  0  0  0  0  0 12
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Last column (k=n) is A000009.
Row sums are A000041.
Row sums without the last column (k=n) are A047967.
For first instead of last part we have A116861, rank A066328.
First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
Using Heinz numbers, this statistic is given by A381437.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
Look-and-Say partition: A048767, A351294, A351295, A381440.
Cf. A047966, A051903, A051904, A091602, A181819, A212166.

egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]

A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 8, 9, 13, 15, 22, 25, 35, 42, 56, 67, 89, 106, 138, 166, 211, 254, 321, 384, 479, 575, 709, 848, 1040, 1239, 1508, 1795, 2168, 2574, 3095, 3661, 4379, 5171, 6154, 7246, 8592, 10088, 11915, 13960, 16425, 19197, 22520, 26253, 30702, 35718

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

			a(1) = 1: [2].
a(2) = 2: [2,2], [2,1,1].
a(3) = 2: [2,2,2], [3,1,1,1].
a(4) = 3: [2,2,2,2], [2,2,1,1,1,1], [4,1,1,1,1].
a(5) = 3: [2,2,2,2,2], [3,2,1,1,1,1,1], [5,1,1,1,1,1].
a(6) = 5: [2,2,2,2,2,2], [2,2,2,1,1,1,1,1,1], [3,3,1,1,1,1,1,1], [4,2,1,1,1,1,1,1], [6,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A002865, A091602, A232623, A332051.

Partial sums give A133041.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
       iquo(n-i*j, i+1))), j=0..min(n/i, k))))
    end:
a:= n-> b(2*n, 1, n)-`if`(n=0, 0, b(2*n, 1, n-1)):
seq(a(n), n=0..60);

CoefficientList[x/(1-x) + (1-x)/QPochhammer[x] + O[x]^60, x] (* Jean-François Alcover, Dec 18 2016 *)

G.f.: x/(1-x) + Product_{k>=2} 1/(1-x^k).
a(0) = 1, a(n) = 1 + A002865(n) = 1 + A000041(n)-A000041(n-1) for n>0.
a(n) = A091602(2n,n) = A096144(2n,n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - Vaclav Kotesovec, Oct 25 2018

A381542 Numbers > 1 whose greatest prime index equals their greatest prime multiplicity.

Original entry on oeis.org

2, 9, 12, 18, 36, 40, 112, 120, 125, 135, 200, 250, 270, 336, 352, 360, 375, 500, 540, 560, 567, 600, 675, 750, 784, 832, 1000, 1008, 1056, 1080, 1125, 1134, 1350, 1500, 1680, 1760, 1800, 2176, 2250, 2268, 2352, 2401, 2464, 2496, 2673, 2700, 2800, 2835, 3000

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   125: {3,3,3}
   135: {2,2,2,3}
   200: {1,1,1,3,3}
   250: {1,3,3,3}
   270: {1,2,2,2,3}
   336: {1,1,1,1,2,4}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Counting partitions by the RHS gives A091602, rank statistic A051903.
For length instead of maximum we have A106529, counted by A047993 (balanced partitions).
For number of distinct factors instead of max index we have A212166, counted by A239964.
Partitions of this type are counted by A240312.
Including number of distinct parts gives A381543, counted by A382302.
A000005 counts divisors.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
A381544 counts partitions without more ones than any other part, ranks A381439.
Cf. A000720, A047966, A048767, A130091, A317090, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A051903(a(n)).

A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 0, 6, 1, 1, 0, 7, 0, 8, 0, 2, 1, 9, 0, 0, 1, 0, 0, 10, 1, 11, 0, 2, 1, 1, 0, 12, 1, 2, 0, 13, 1, 14, 0, 0, 1, 15, 0, 0, 0, 2, 0, 16, 0, 2, 0, 2, 1, 17, 0, 18, 1, 0, 0, 3, 1, 19, 0, 2, 1, 20, 0, 21, 1, 0, 0, 1, 1, 22, 0, 0, 1, 23

Offset: 2

Gus Wiseman, Jul 10 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.

			The 0-prepended prime indices of 9842 are {0,1,4,8,12}, with differences (1,3,4,4), so a(9842) = 1.

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

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of positive terms are A005117, complement A013929.
A similar statistic is counted by A238353.
The maximal version is A286469, without prepending A355526.
Without prepending we have A355524 or A355525.
Positions of ones are A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A112798 lists prime indices, with sum A056239.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A064428, A066312, A091602, A120944, A238354, A286470, A325161, A352822, A355527, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min@@Differences[Prepend[primeMS[n],0]],{n,2,100}]

cp's OEIS Frontend

A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.

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A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

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A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

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A213177 Number T(n,k) of parts in all partitions of n with largest multiplicity k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

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A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

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A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

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A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.

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