A239964 -id:A239964 - cp's OEIS frontend

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

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}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
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.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

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

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

A386638 Number of integer partitions of n of inseparable type.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296

Offset: 0

Gus Wiseman, Aug 14 2025

nonn

A multiset is inseparable iff it has no permutation without adjacent equal parts. It is of inseparable type iff any multiset with those multiplicities (type) is inseparable. For example, {1,1,2} is separable but {1,1,1,2} is not; hence (2,1) is of separable type but (3,1) is not.

Also the number of integer partitions of n whose greatest part is at least two more than the sum of all the other parts.

			The a(2) = 1 through a(10) = 12 partitions (A=10):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)
            (31)  (41)  (42)   (52)   (53)    (63)    (64)
                        (51)   (61)   (62)    (72)    (73)
                        (411)  (511)  (71)    (81)    (82)
                                      (521)   (621)   (91)
                                      (611)   (711)   (622)
                                      (5111)  (6111)  (631)
                                                      (721)
                                                      (811)
                                                      (6211)
                                                      (7111)
                                                      (61111)

Reduplication of A000070 shifted right.
Same as A025065 shifted right twice.
The Heinz numbers of these partitions are A335126.
Row sums of A386586.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, inseparable case A386632.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A336106 counts separable type partitions, ranks A335127, sums of A386585.
A386633 counts separable type set partitions, row sums of A386635.
A386634 counts inseparable type set partitions, row sums of A386636.
Cf. A000110, A008284, A047966, A072233, A106351, A124762, A217605, A239964, A279790, A335125, A351293, A386577.

Table[Length[Select[IntegerPartitions[n],2*Max@@#>1+n&]],{n,0,15}]

For n>1, a(n) = A025065(n-2).

a(n) = A000041(n) - A336106(n).

A240306 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (number of distinct parts of p).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 76, 100, 131, 169, 212, 278, 352, 442, 555, 703, 871, 1088, 1342, 1664, 2046, 2517, 3064, 3758, 4574, 5548, 6718, 8119, 9797, 11784, 14150, 16935, 20263, 24179, 28798, 34237, 40677, 48122, 57008, 67291, 79400

Offset: 0

Clark Kimberling, Apr 05 2014

			a(6) counts these 6 partitions:  6, 51, 42, 411, 321, 2211.

Cf. A240305, A239964, A240308, A240309, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

a(n) = A240305(n) + A239964(n) for n >= 0.

a(n) + A240308(n) = A000041(n) for n >= 0.

A240309 Number of partitions p of n such that (maximal multiplicity of the parts of p) > (number of distinct parts of p).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 59, 76, 100, 128, 173, 212, 275, 350, 447, 552, 704, 870, 1094, 1346, 1672, 2048, 2540, 3084, 3775, 4595, 5592, 6764, 8180, 9853, 11865, 14250, 17075, 20404, 24376, 29024, 34498, 41012, 48550, 57463, 67873

Offset: 0

Clark Kimberling, Apr 05 2014

			a(6) counts these 5 partitions:  33, 3111, 222, 21111, 111111.

Cf. A240305, A240306, A239964, A240308, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

a(n) = A240308(n) - A239964(n) for n >= 0.

a(n) + A240305(n) + A239964(n) = A000041(n) for n >= 0.

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A240305 Number of partitions p of n such that (maximal multiplicity of the parts of p) < (number of distinct parts of p).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 5, 7, 12, 14, 21, 29, 38, 50, 70, 90, 117, 156, 196, 253, 324, 411, 514, 650, 809, 1015, 1259, 1555, 1917, 2365, 2898, 3536, 4318, 5248, 6365, 7691, 9297, 11180, 13446, 16115, 19296, 23019, 27474, 32653, 38838, 46002, 54511, 64371, 76012

Offset: 0

Clark Kimberling, Apr 05 2014

			a(6) counts these 3 partitions:  51, 42, 321.

Cf. A240306, A239964, A240308, A240309, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

a(n) = A240306(n) - A239964(n) for n >= 0.

a(n) + A239964(n) + A240309(n) = A000041(n) for n >= 0.

A240308 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 8, 10, 15, 18, 28, 35, 48, 63, 85, 106, 141, 180, 229, 294, 374, 468, 591, 741, 925, 1149, 1421, 1751, 2163, 2648, 3239, 3944, 4813, 5825, 7062, 8518, 10286, 12340, 14835, 17739, 21223, 25287, 30155, 35787, 42522, 50296, 59556, 70243, 82902

Offset: 0

Clark Kimberling, Apr 05 2014

			a(6) counts these 8 partitions:  6, 411, 33, 3111, 222, 2211, 21111, 111111.

Cf. A240305, A240306, A239964, A240309, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

a(n) = A239964(n) + A240308(n) for n >= 0.

a(n) + A240305(n) = A000041(n) for n >= 0.

A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

Original entry on oeis.org

6, 1, 8, 32, 64, 128, 256, 6144, 512, 27648, 1024, 73728, 2048, 147456, 165888, 4096, 248832, 196608, 8192, 497664, 1119744, 393216, 16384, 2239488

Offset: 0

Gus Wiseman, Apr 11 2025

Also the position of first appearance of n in A382525 (number of times n appears in A048767).
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. 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)).
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:
       6: {1,2}
       1: {}
       8: {1,1,1}
      32: {1,1,1,1,1}
      64: {1,1,1,1,1,1}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
    6144: {1,1,1,1,1,1,1,1,1,1,1,2}
     512: {1,1,1,1,1,1,1,1,1}
   27648: {1,1,1,1,1,1,1,1,1,1,2,2,2}
    1024: {1,1,1,1,1,1,1,1,1,1}
   73728: {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
  147456: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
  165888: {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  248832: {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}

Positions of first appearances in A382525.
The Look-and-Say partition is ranked by A048767, listed by A381440.
Look-and-Say partitions are counted by A239455, complement A351293.
Look-and-Say partitions are ranked by A351294.
Non-Look-and-Say partitions are ranked by A351295, conjugate A381433.
The section-sum partition is ranked by A381431, listed by A381436.
Section-sum partitions are ranked by A381432.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Cf. A003557, A047966, A048768, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964, A381540.

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#], UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
z=Table[Length[stp[Last/@FactorInteger[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[z,k][[1,1]],{k,0,mnrm[z+1]-1}]

A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

Original entry on oeis.org

2, 9, 24, 54, 72, 80, 108, 125, 216, 224, 400, 704, 960, 1215, 1250, 1568, 1664, 2000, 2401, 2500, 2688, 2880, 4352, 4800, 5000, 5103, 6075, 7290, 7744, 8064, 8448, 8640, 8960, 9375, 9728, 10000, 10976, 14400, 14580, 18816, 19968, 21632, 23552, 24000, 24057

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}
     24: {1,1,1,2}
     54: {1,2,2,2}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    108: {1,1,2,2,2}
    125: {3,3,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    400: {1,1,1,1,3,3}
    704: {1,1,1,1,1,1,5}
    960: {1,1,1,1,1,1,2,3}

For (length) instead of (sum of distinct) we have A000961.
Including number of parts gives A062457 (degenerate).
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
Partitions of this type are counted by A381079.
A001222 counts prime factors, distinct A001221.
A047993 counts partitions with max part = length, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, complement A351293.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Last/@FactorInteger[#]==Total[Union[prix[#]]]&]

A051903(a(n)) = A066328(a(n)).

A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 8, 6, 15, 13, 19, 25, 33, 36, 54, 58, 80, 96, 122, 141, 188, 217, 274, 326, 408, 474, 600, 695, 859, 1012, 1233, 1440, 1763, 2050, 2475, 2899, 3476, 4045, 4850, 5630, 6695, 7797, 9216, 10689, 12628, 14611, 17162, 19875, 23253

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(3) = 1 through a(10) = 8 partitions:
  (21)  (31)  (41)  (51)    (61)   (71)    (81)      (91)
                    (321)   (421)  (431)   (531)     (541)
                    (2211)         (521)   (621)     (631)
                                   (3311)  (32211)   (721)
                                           (222111)  (4321)
                                                     (4411)
                                                     (33211)
                                                     (42211)

First differences of A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A360014.
Equal case of A381544 (ranks A381439).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A051904, A091605, A116861, A237984, A239455, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]==Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

cp's OEIS Frontend

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

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A386638 Number of integer partitions of n of inseparable type.

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A240306 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (number of distinct parts of p).

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A240309 Number of partitions p of n such that (maximal multiplicity of the parts of p) > (number of distinct parts of p).

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A381544 Number of integer partitions of n not containing more ones than any other part.

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A240305 Number of partitions p of n such that (maximal multiplicity of the parts of p) < (number of distinct parts of p).

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A240308 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).

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A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

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A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

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