A363723 -id:A363723 - cp's OEIS frontend

A363953 Number of integer partitions of n with high mode k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 2, 2, 1, 1, 1, 0, 7, 2, 1, 2, 1, 1, 1, 0, 9, 4, 2, 2, 2, 1, 1, 1, 0, 13, 6, 2, 2, 2, 2, 1, 1, 1, 0, 18, 7, 4, 3, 3, 2, 2, 1, 1, 1, 0, 26, 10, 5, 2, 3, 3, 2, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Jul 07 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

Extending the terminology of A124944, the "high mode" in a multiset is the greatest mode.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  3  1  1  1  1
  0  4  2  2  1  1  1
  0  7  2  1  2  1  1  1
  0  9  4  2  2  2  1  1  1
  0 13  6  2  2  2  2  1  1  1
  0 18  7  4  3  3  2  2  1  1  1
  0 26 10  5  2  3  3  2  2  1  1  1
  0 32 15  8  4  4  4  3  2  2  1  1  1
  0 47 19  9  5  3  4  4  3  2  2  1  1  1
  0 60 26 13  7  5  5  5  4  3  2  2  1  1  1
  0 79 34 18 10  6  5  5  5  4  3  2  2  1  1  1
Row n = 9 counts the following partitions:
  .  (711)        (522)     (333)   (441)  (54)   (63)   (72)  (81)  (9)
     (6111)       (4221)    (3321)  (432)  (531)  (621)
     (5211)       (3222)
     (51111)      (32211)
     (4311)       (22221)
     (42111)      (222111)
     (411111)
     (33111)
     (321111)
     (3111111)
     (2211111)
     (21111111)
     (111111111)

Row sums are A000041.
For median: A124944 (low A124943), rank statistic A363942 (low A363941).
Column k = 1 is A241131 (partitions w/ high mode 1), ranks A360013, A360015.
The rank statistic for this triangle is A363487, low A363486.
For mean: A363946 (low A363945), rank statistic A363944 (low A363943).
The low version is A363952.
A008284 counts partitions by length, A058398 by mean.
A362612 counts partitions (max part) = (unique mode), ranks A362616.
A362614 counts partitions by number of modes, rank statistic A362611.
A362615 counts partitions by number of co-modes, rank statistic A362613.
Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.
Cf. A002865, A025065, A026905, A237984, A363723, A363724, A363731.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], If[Length[#]==0,0,Last[modes[#]]]==k&]],{n,0,15},{k,0,n}]

A364062 Number of integer partitions of n with unique co-mode 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 6, 2, 8, 6, 9, 6, 16, 7, 21, 12, 23, 18, 39, 17, 47, 32, 59, 40, 86, 44, 110, 72, 131, 95, 188, 103, 233, 166, 288, 201, 389, 244, 490, 347, 587, 440, 794, 524, 974, 727, 1187, 903, 1547, 1106, 1908, 1459, 2303, 1826, 2979, 2198

Offset: 0

Gus Wiseman, Jul 12 2023

nonn

These are partitions with at least one 1 but with fewer 1's than each of the other parts.

We define a co-mode in a multiset to be an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			The a(n) partitions for n = 5, 7, 11, 13, 15:
  (221)    (331)      (551)          (661)            (771)
  (11111)  (2221)     (33221)        (4441)           (44331)
           (1111111)  (33311)        (33331)          (55221)
                      (222221)       (44221)          (442221)
                      (2222111)      (332221)         (3322221)
                      (11111111111)  (2222221)        (3333111)
                                     (22222111)       (22222221)
                                     (1111111111111)  (222222111)
                                                      (111111111111111)

For high (or unique) mode we have A241131, ranks A360013.
For low mode we have A241131, ranks A360015.
Allowing any unique co-mode gives A362610, ranks A359178.
These partitions have ranks A364061.
Adding all 1-free partitions gives A364159, ranks A364158.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A237984 counts partitions containing their mean, ranks A327473.
A327472 counts partitions not containing their mean, ranks A327476.
A362608 counts partitions w/ unique mode, ranks A356862, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Cf. A002865, A027336, A325347, A098859, A360014, A362562, A362606, A363720, A363723, A363725, A363726.

comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],comodes[#]=={1}&]],{n,0,30}]

A362562 Number of non-constant integer partitions of n having a unique mode equal to the mean.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 52, 12, 14, 33, 54, 0, 121, 0, 98, 76, 31, 100, 343, 0, 45, 164, 493, 0, 548, 0, 483, 757, 88, 0, 1789, 289, 979, 645, 1290, 0, 2225, 1677, 3371, 1200, 221, 0, 10649

Offset: 0

Gus Wiseman, Jun 27 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(8) = 1 through a(16) = 7 partitions:
  (3221)  .  (32221)  .  (4332)    .  (3222221)  (43332)  (5443)
                         (5331)       (3322211)  (53331)  (6442)
                         (322221)     (4222211)  (63321)  (7441)
                         (422211)                         (32222221)
                                                          (33222211)
                                                          (42222211)
                                                          (52222111)

Partitions containing their mean are counted by A237984, ranks A327473.
Partitions missing their mean are counted by A327472, ranks A327476.
Allowing constant partitions gives A363723.
Including median also gives A363728, ranks A363729.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode.
Cf. A240219, A325347, A363719, A363720, A363724, A363725, A363731, A363740.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}==modes[#]&]],{n,0,30}]

A363722 Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 90, 121, 125, 128, 169, 243, 256, 270, 289, 343, 361, 512, 525, 529, 550, 625, 729, 756, 810, 841, 961, 1024, 1331, 1369, 1666, 1681, 1849, 1911, 1950, 2048, 2187, 2197, 2209, 2268, 2401, 2430, 2625, 2695, 2700, 2750, 2809

Offset: 1

Gus Wiseman, Jun 24 2023

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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    90: {1,2,2,3}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}

These partitions are counted by A363719 - 1 for n > 0.
Including primes gives A363727, counted by A363719.
For prime powers instead of just primes we have A363729, counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A000961, A327473, A327476, A359893, A359908, A360009, A360550, A363721, A363725, A363741.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],!PrimeQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

Complement of A000040 in A363727.

Assuming there is a unique mode, we have A326567(a(n))/A326568(a(n)) = A360005(a(n))/2 = A363486(a(n)) = A363487(a(n)).

A363745 Number of integer partitions of n whose rounded-down mean is 2.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 4, 10, 6, 16, 21, 24, 32, 58, 47, 85, 111, 119, 158, 248, 217, 341, 442, 461, 596, 867, 792, 1151, 1465, 1506, 1916, 2652, 2477, 3423, 4298, 4381, 5488, 7334, 6956, 9280, 11503, 11663, 14429, 18781, 17992, 23383, 28675, 28970, 35449, 45203

Offset: 0

Gus Wiseman, Jul 05 2023

nonn

			The a(2) = 1 through a(10) = 16 partitions:
  (2)  .  (22)  (32)  (222)  (322)  (332)   (3222)  (3322)
          (31)  (41)  (321)  (331)  (422)   (3321)  (3331)
                      (411)  (421)  (431)   (4221)  (4222)
                             (511)  (521)   (4311)  (4321)
                                    (611)   (5211)  (4411)
                                    (2222)  (6111)  (5221)
                                    (3221)          (5311)
                                    (3311)          (6211)
                                    (4211)          (7111)
                                    (5111)          (22222)
                                                    (32221)
                                                    (33211)
                                                    (42211)
                                                    (43111)
                                                    (52111)
                                                    (61111)

For 1 instead of 2 we have A025065, ranks A363949.
The high version is A026905 reduplicated, ranks A363950.
Column k = 2 of A363945.
These partitions have ranks A363954.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A027336, A237984, A241131, A327472, A327482, A363723, A363943, A363944, A363946.

Table[Length[Select[IntegerPartitions[n],Floor[Mean[#]]==2&]],{n,0,30}]

A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 11, 18, 26, 35, 49, 70, 89, 123, 164, 212, 278, 366, 460, 597, 762, 957, 1210, 1530, 1891, 2369, 2943, 3621, 4468, 5507, 6703, 8210, 10004, 12115, 14688, 17782, 21365, 25743, 30913, 36965, 44210, 52801, 62753, 74667, 88626, 104874, 124070

Offset: 0

Gus Wiseman, Jul 06 2023

nonn

We use the "rounding half to even" rule, see link.

			The a(0) = 0 through a(8) = 18 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)
             (21)  (22)  (32)   (33)    (43)    (44)
                   (31)  (41)   (42)    (52)    (53)
                         (221)  (51)    (61)    (62)
                         (311)  (222)   (322)   (71)
                                (321)   (331)   (332)
                                (411)   (421)   (422)
                                (2211)  (511)   (431)
                                (3111)  (2221)  (521)
                                        (3211)  (611)
                                        (4111)  (2222)
                                                (3221)
                                                (3311)
                                                (4211)
                                                (5111)
                                                (22211)
                                                (32111)
                                                (41111)

Wikipedia, Rounding.

Rounding-up gives A000065.
Rounding-down gives A110618, ranks A344291.
For median instead of mean we appear to have A238495.
The complement is counted by A363947, ranks A363948.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A067538 counts partitions with integer mean, ranks A316413.
A124943 counts partitions by low median, high A124944.
Cf. A002865, A098859, A241131, A327482, A363723, A363724, A363731, A363946.

Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]>1&]],{n,0,30}]

a(n) = A000041(n) - A363947(n).

A364193 Number of integer partitions of n where the least part is the unique mode.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 7, 9, 13, 17, 24, 32, 43, 58, 75, 97, 130, 167, 212, 274, 346, 438, 556, 695, 865, 1082, 1342, 1655, 2041, 2511, 3067, 3756, 4568, 5548, 6728, 8130, 9799, 11810, 14170, 16980, 20305, 24251, 28876, 34366, 40781, 48342, 57206, 67597, 79703

Offset: 0

Gus Wiseman, Jul 16 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (3211)     (611)
                                     (3111)    (4111)     (2222)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For greatest part and multiple modes we have A171979.
Allowing multiple modes gives A240303.
For greatest instead of least part we have A362612, ranks A362616.
For mean instead of least part we have A363723.
These partitions have ranks A364160.
A000041 counts integer partitions.
A362611 counts modes in prime factorization, A362613 co-modes.
A362614 counts partitions by number of modes, co-modes A362615.
A363486 gives least mode in prime indices, A363487 greatest.
A363952 counts partitions by low mode, A363953 high.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A327472, A360015.

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Last[Length/@Split[#]]>Max@@Most[Length/@Split[#]]&]]],{n,0,30}]

A363265 Number of integer factorizations of n with a unique mode.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 27 2023

nonn

An integer factorization of n is a multiset of positive integers > 1 with product n.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Conjecture: 9 is missing from this sequence.

			The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
  (2)  (4)    (16)       (24)       (48)         (72)
       (2*2)  (4*4)      (2*2*6)    (3*4*4)      (2*6*6)
              (2*2*4)    (2*2*2*3)  (2*2*12)     (3*3*8)
              (2*2*2*2)             (2*2*2*6)    (2*2*18)
                                    (2*2*3*4)    (2*2*2*9)
                                    (2*2*2*2*3)  (2*2*3*6)
                                                 (2*3*3*4)
                                                 (2*2*2*3*3)

The complement for partitions is A362607, ranks A362605.
The version for partitions is A362608, ranks A356862.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A339846 counts even-length factorizations, A339890 odd-length.
Cf. A240219, A326622, A333487, A335434, A347438, A362610, A362611, A362612, A362614, A363723.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n],Length[modes[#]]==1&]],{n,100}]

A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 2, 4, 0, 0, 1, 0, 2, 5, 3, 0, 0, 1, 0, 4, 7, 0, 3, 0, 0, 1, 0, 4, 8, 5, 4, 0, 0, 0, 1, 0, 4, 14, 7, 4, 0, 0, 0, 0, 1, 0, 7, 21, 8, 0, 5, 0, 0, 0, 0, 1, 0, 7, 22, 11, 10, 0, 5, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jul 07 2023

We use the "rounding half to even" rule, see link.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  2  0  1
  0  2  4  0  0  1
  0  2  5  3  0  0  1
  0  4  7  0  3  0  0  1
  0  4  8  5  4  0  0  0  1
  0  4 14  7  4  0  0  0  0  1
  0  7 21  8  0  5  0  0  0  0  1
  0  7 22 11 10  0  5  0  0  0  0  1
  0  7 36 15 12  0  6  0  0  0  0  0  1
  0 12 32 36 14  0  6  0  0  0  0  0  0  1
  0 12 53 23 23 16  0  7  0  0  0  0  0  0  1
  0 12 80 30 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (31111)    (511)   .  (61)  .  .  (7)
     (22111)    (421)      (52)
     (211111)   (4111)     (43)
     (1111111)  (331)
                (322)
                (3211)
                (2221)

Wikipedia, Rounding.

Row sums are A000041.
The rank statistic for this triangle is A363489.
The version for low mean is A363945, rank statistic A363943.
The version for high mean is A363946, rank statistic A363944.
Column k = 1 is A363947 (A026905 tripled).
A008284 counts partitions by length, A058398 by mean.
A026905 redoubled counts partitions with high mean 2, ranks A363950.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
More triangles: A124943, A124944, A363952, A363953.
Cf. A002865, A025065, A237984, A327472, A327482, A363723, A363724, A363731.

Table[If[n==k==0,1,Length[Select[IntegerPartitions[n], Round[Mean[#]]==k&]]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A363953 Number of integer partitions of n with high mode k.

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A364062 Number of integer partitions of n with unique co-mode 1.

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A362562 Number of non-constant integer partitions of n having a unique mode equal to the mean.

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A363722 Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A363745 Number of integer partitions of n whose rounded-down mean is 2.

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A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

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A364193 Number of integer partitions of n where the least part is the unique mode.

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A363265 Number of integer factorizations of n with a unique mode.

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A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

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