A067538 -id:A067538 - cp's OEIS frontend

A231429 Number of partitions of 2n into distinct parts < n.

1, 0, 0, 0, 0, 1, 2, 4, 8, 14, 22, 35, 53, 78, 113, 160, 222, 306, 416, 558, 743, 980, 1281, 1665, 2149, 2755, 3514, 4458, 5626, 7070, 8846, 11020, 13680, 16920, 20852, 25618, 31375, 38309, 46649, 56651, 68616, 82908, 99940, 120192, 144238, 172730, 206425

Offset: 0

Reinhard Zumkeller, Nov 14 2013

nonn

From Gus Wiseman, Jun 17 2023: (Start)
Also the number of integer compositions of n with weighted sum 3*n, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i. The a(0) = 1 through a(9) = 14 compositions are:
  ()  .  .  .  .  (11111)  (3111)   (3211)   (3311)    (3411)
                           (11211)  (11311)  (4121)    (4221)
                                    (12121)  (11411)   (5112)
                                    (21112)  (12221)   (11511)
                                             (13112)   (12321)
                                             (21131)   (13131)
                                             (21212)   (13212)
                                             (111122)  (21231)
                                                       (21312)
                                                       (22122)
                                                       (31113)
                                                       (111141)
                                                       (111222)
                                                       (112113)
For partitions we have A363527, ranks A363531. For reversed partitions we have A363526, ranks A363530.
(End)

			a(5) = #{4+3+2+1} = 1;
a(6) = #{5+4+3, 5+4+2+1} = 2;
a(7) = #{6+5+3, 6+5+2+1, 6+4+3+1, 5+4+3+2} = 4;
a(8) = #{7+6+3, 7+6+2+1, 7+6+3, 7+5+3+1, 7+4+3+2, 6+5+4+1, 6+5+3+2, 6+4+3+2+1} = 8;
a(9) = #{8+7+3, 8+7+2+1, 8+6+4, 8+6+3+1, 8+5+4+1, 8+5+3+2, 8+4+3+2+1, 7+6+5, 7+6+4+1, 7+6+3+2, 7+5+4+2, 7+5+3+2+1, 6+5+4+3, 6+5+4+2+1} = 14.

Cf. A209815, A079122.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A008284, A029931, A067538, A222855, A222955, A222970, A359042, A360672, A360675, A362559.

a231429 n = p [1..n-1] (2*n) where
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Total[Accumulate[#]]==3n&]],{n,0,15}] (* Gus Wiseman, Jun 17 2023 *)

A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 4, 3, 4, 4, 8, 5, 7, 8, 10, 8, 13, 13, 20, 18, 25, 25, 36, 34, 48, 52, 64, 64, 85, 85, 108, 106, 129, 133, 160, 158, 189, 194, 229, 228, 276, 279, 332, 336, 394, 402, 476, 489, 572, 599, 699, 728, 845, 889, 1032, 1094, 1251, 1332, 1523

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

			The a(13) = 8 partitions are (4441), (55111), (322222), (332221), (333211), (622111), (631111), (7111111).

Cf. A067538, A074761, A143773, A285572, A289508, A290103, A305566, A316429, A316430, A316431, A316432.

Table[Length[Select[IntegerPartitions[n],LCM@@#==Length[#]&]],{n,30}]

a(n) = {my(nb = 0); forpart(p=n, if (lcm(Vec(p))==#p, nb++);); nb;} \\ Michel Marcus, Jul 03 2018

A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers are in the sequence begins: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (3,2,1), (11), (12).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319334.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],LCM@@primeMS[#]==Total[primeMS[#]]&]

A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 6, 1, 3, 1, 2, 1, 1, 1, 3, 1, 6, 1, 5, 1, 2, 2, 2, 4, 3, 1, 9, 1, 1, 3, 1, 1, 4, 1, 4, 2, 6, 1, 6, 1, 3, 7, 4, 2, 5, 1, 10, 1, 3, 1, 9, 3

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(55) = 2 through a(60) = 9 partitions:
  (55)           (56)         (57)        (58)    (59)  (60)
  (27,16,9,2,1)  (24,18,8,6)  (49,7,1)    (49,9)        (54,6)
                              (27,25,5)   (50,8)        (48,12)
                              (27,18,12)                (27,24,9)
                                                        (27,24,6,2,1)
                                                        (36,12,9,2,1)
                                                        (36,9,6,4,3,2)
                                                        (24,18,9,6,2,1)
                                                        (27,16,9,4,3,1)

Wikipedia, Geometric mean

Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean are A102627.
Strict partitions with integer geometric mean are A326625.
Non-constant partitions with integer mean and geometric mean are A326641.
Subsets with integer mean and geometric mean are A326643.
Heinz numbers of partitions with integer mean and geometric mean are A326645.
Cf. A051293, A067538, A067539, A078175, A082553, A316413, A326027, A326623, A326644, A326646, A326647.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,30}]

More terms from Jinyuan Wang, Jun 26 2020

A329976 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) > (number of numbers in p having multiplicity > 1).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 6, 9, 14, 18, 27, 38, 50, 66, 89, 113, 145, 186, 234, 297, 374, 468, 585, 737, 912, 1140, 1407, 1758, 2153, 2668, 3254, 4007, 4855, 5946, 7170, 8705, 10451, 12626, 15068, 18125, 21551, 25766, 30546, 36365, 42958, 50976, 60062, 70987

Offset: 0

Clark Kimberling, Feb 03 2020

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d > r.
Also the number of integer partitions of n with median multiplicity 1. - Gus Wiseman, Mar 20 2023

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 4.

For parts instead of multiplicities we have A027336
The complement is counted by A330001.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A237363 counts partitions with median difference 0.
Cf. A000975, A027193, A067538, A240219, A241274, A325347, A359893, A360005.

z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] > r[p]], {n, 0, z}]

a(n) + A241274(n) + A330001(n) = A000041(n) for n >= 0.

A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose number of factors is divisible by their least common multiple.

			The a(n) factorizations for n = 8192, 46656, 73728:
  2*2*2*2*2*4*8*8          6*6*6*6*6*6              2*2*2*2*2*2*2*2*2*4*6*6
  2*2*2*2*4*4*4*8          2*2*2*2*2*2*3*3*3*3*3*3  2*2*2*2*2*2*2*2*3*4*4*6
  2*2*2*4*4*4*4*4                                   2*2*2*2*2*2*2*3*3*4*4*4
  2*2*2*2*2*2*2*2*2*2*2*4                           2*2*2*2*2*2*2*2*2*2*6*12
                                                    2*2*2*2*2*2*2*2*2*3*4*12

The version for partitions is A340693, with reciprocal version A143773.
Positions of nonzero terms are A340852.
The reciprocal version is A340853.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even numbers, even-length case A340786.
A340831/A340832 count factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],And@@IntegerQ/@(Length[#]/#)&]],{n,100}]

A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose greatest common divisor is a multiple of the number of factors.

			The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
  2   4     16    48     96     144     216      240     432
      2*2   2*8   6*8    2*48   2*72    4*54     4*60    6*72
            4*4   2*24   4*24   4*36    6*36     6*40    8*54
                  4*12   6*16   6*24    12*18    8*30    12*36
                         8*12   8*18    2*108    10*24   18*24
                                12*12   6*6*6    12*20   2*216
                                        3*3*24   2*120   4*108
                                        3*6*12           3*3*48
                                                         3*6*24
                                                         6*6*12
                                                         3*12*12

Positions of 1's are A048103.
Positions of terms > 1 are A100716.
The version for partitions is A143773 (A316428).
The reciprocal for partitions is A340693 (A340606).
The version for strict partitions is A340830.
The reciprocal version is A340851.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even factors, even-length case A340786.
A340831/A340832 counts factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],n>1&&Divisible[GCD@@#,Length[#]]&]],{n,100}]

A361852 Number of integer partitions of n such that (length) * (maximum) < 2n.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 17, 21, 27, 37, 41, 58, 67, 80, 106, 126, 153, 193, 209, 263, 326, 402, 419, 565, 650, 694, 891, 1088, 1120, 1419, 1672, 1987, 2245, 2345, 2856, 3659, 3924, 4519, 4975, 6407, 6534, 8124, 8280, 9545, 12937, 13269, 13788, 16474, 20336

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also partitions such that (maximum) < 2*(mean).

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (2211)    (421)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (22111)
                                               (211111)
                                               (1111111)
For example, the partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 < 2*7, so y is counted under a(7).

For length instead of mean we have A237754.
Allowing equality gives A237755, for median A361848.
For equal median we have A361849, ranks A361856.
The equal version is A361853, ranks A361855.
For median instead of mean we have A361858.
The complement is counted by A361906.
Reversing the inequality gives A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A027193, A111907, A116608, A237824, A237984, A324517, A327482, A349156, A360068, A360071, A361394.

Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#<2n&]],{n,30}]

A361866 Number of set partitions of {1..n} with block-means summing to an integer.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 75, 267, 1119, 4965, 22694, 117090, 670621, 3866503, 24113829, 161085223, 1120025702, 8121648620, 62083083115, 492273775141, 4074919882483

Offset: 0

Gus Wiseman, Apr 04 2023

			The a(1) = 1 through a(4) = 8 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{13}{24}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition y = {{1,2},{3,4}} has block-means {3/2,7/2}, with sum 5, so y is counted under a(4).

For mean instead of sum we have A361865, for median A361864.
For median instead of mean we have A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions with integer mean block-size.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean, median A013580.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A067659, A326515, A326516, A326521.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Total[Mean/@#]]&]],{n,6}]

a(14)-a(20) from Christian Sievers, May 12 2025

A363622 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with weighted alternating sum k (leading and trailing 0's omitted).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 1, 3, 0, 0, 2, 0, 1, 1, 2, 1, 1, 3, 0, 2, 2, 1, 1, 2, 2, 1, 1, 5, 0, 0, 3, 0, 2, 2, 2, 1, 3, 2, 1, 1, 5, 0, 3, 3, 2, 2, 3, 2, 2, 4, 2, 1, 1, 7, 0, 0, 5, 0, 3, 3, 4, 2, 4, 2, 4, 4, 2, 1, 1

Offset: 0

Gus Wiseman, Jun 15 2023

We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) i * y_i. For example:
- (3,3,2,1,1) has weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 = 4.
- (1,2,2,3) has weighted alternating sum  1*1 - 2*2 + 3*2 - 4*3 = -9.

			Triangle begins:
  1
  1
  1  0  0  1
  1  0  1  1
  2  0  0  1  0  1  1
  2  0  1  1  1  1  1
  3  0  0  2  0  1  1  2  1  1
  3  0  2  2  1  1  2  2  1  1
  5  0  0  3  0  2  2  2  1  3  2  1  1
  5  0  3  3  2  2  3  2  2  4  2  1  1
  7  0  0  5  0  3  3  4  2  4  2  4  4  2  1  1
  7  0  5  5  3  3  5  4  3  5  3  5  4  2  1  1
Row n = 6 counts the following partitions:
  k=-3            k=0        k=2    k=3   k=4      k=5    k=6
  -----------------------------------------------------------
  (33)      .  .  (42)    .  (321)  (51)  (222)    (411)  (6)
  (2211)          (3111)                  (21111)
  (111111)

Row sums are A000041.
The unweighted version is A103919 with leading zeros removed.
Row-lengths appear to be A168233.
Central column T(n,0) is A363532, ranks A363621.
The corresponding rank statistic is A363619, reverse A363620.
The reverse version is A363623.
A053632 counts compositions by weighted sum.
A264034 counts partitions by weighted sum, reverse A358194.
A316524 gives alternating sum of prime indices, reverse A344616.
A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
Cf. A008284, A067538, A222855, A222970, A318283, A320387, A360672, A360675, A362559, A363626.

altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
Table[Length[Select[IntegerPartitions[n],altwtsum[#]==k&]],{n,0,15},{k,Min[altwtsum/@IntegerPartitions[n]], Max[altwtsum/@IntegerPartitions[n]]}]

cp's OEIS Frontend

A231429 Number of partitions of 2n into distinct parts < n.

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A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

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A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

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A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

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A329976 Number of partitions p of n such that (number of numbers in p that have multiplicity 1) > (number of numbers in p having multiplicity > 1).

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Formula

A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

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A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

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A361852 Number of integer partitions of n such that (length) * (maximum) < 2n.

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A361866 Number of set partitions of {1..n} with block-means summing to an integer.

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