A244515 -id:A244515 - cp's OEIS frontend

A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 19, 15, 26, 1, 13, 39, 4, 25, 47, 5, 19, 70, 12, 29, 89, 17, 33, 115, 28, 42, 148, 41, 39, 189, 69, 62, 235, 88, 55, 294, 141, 81, 362, 183, 1, 84, 450, 253, 5, 103, 558, 333, 8, 105, 669, 464, 17, 153, 817, 576, 29

Offset: 1

Gus Wiseman, Apr 15 2019

For example, the partition (32111) has multiplicities {1,1,3}, of which 2 are distinct, so is counted under T(8,2).

			Triangle begins:
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  19
  15  26   1
  13  39   4
  25  47   5
  19  70  12
  29  89  17
  33 115  28
  42 148  41
  39 189  69
  62 235  88
  55 294 141
  81 362 183   1
Row n = 8 counts the following partitions:
  (8)         (332)
  (44)        (422)
  (53)        (611)
  (62)        (3221)
  (71)        (4211)
  (431)       (5111)
  (521)       (22211)
  (2222)      (32111)
  (3311)      (41111)
  (11111111)  (221111)
              (311111)
              (2111111)

Alois P. Heinz, Rows n = 1..200, flattened

Row lengths are A056556. Row sums are A000041. Column k = 1 is A047966. Column k = 2 is A325243.

Cf. A008284, A062770, A071625, A098859, A116608, A127002, A183558, A243978, A244515, A325244, A325268.

DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==k&]],{n,20},{k,n}],0,2]

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.

Original entry on oeis.org

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 *)

A325240 Numbers whose minimum prime exponent is 2.

Original entry on oeis.org

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

Or barely powerful numbers, a subset of powerful numbers A001694.
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 whose minimum multiplicity is 2 (counted by A244515).
Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

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

Positions of 2's in A051904.
Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Cf. A065483, A082695.

Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023

from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

A325243 Number of integer partitions of n with exactly two distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 19, 26, 39, 47, 70, 89, 115, 148, 189, 235, 294, 362, 450, 558, 669, 817, 980, 1197, 1421, 1709, 2012, 2429, 2836, 3380, 3961, 4699, 5433, 6457, 7433, 8770, 10109, 11818, 13547, 15912, 18109, 21105, 24121, 27959, 31736, 36840, 41670

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1 and 2) so is counted under a(9).

The Heinz numbers of these partitions are given by A323055.

			The a(4) = 1 through a(9) = 19 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

Column k = 2 of A325242. Dominated by A325267.

Cf. A008284, A062770, A071625, A098859, A116608, A244515, A323055, A325244.

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

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A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

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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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A325240 Numbers whose minimum prime exponent is 2.

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A325243 Number of integer partitions of n with exactly two distinct multiplicities.

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