A356862 -id:A356862 - cp's OEIS frontend

A362609 Number of integer partitions of n with more than one part of least multiplicity.

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 19, 26, 42, 51, 74, 103, 136, 174, 246, 303, 411, 523, 674, 844, 1114, 1364, 1748, 2174, 2738, 3354, 4247, 5139, 6413, 7813, 9613, 11630, 14328, 17169, 20958, 25180, 30497, 36401, 44025, 52285, 62834, 74626, 89111, 105374, 125662

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

These are partitions where no part appears fewer times than all of the others.

			The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1 (namely 1 and 4), so is counted under a(9).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (42111)
                                             (222111)
                                             (321111)

For parts instead of multiplicities we have A117989, ranks A283050.
For median instead of co-mode we have A238479, complement A238478.
These partitions have ranks A362606.
For mode instead of co-mode we have A362607, ranks A362605.
For mode complement instead of co-mode we have A362608, ranks A356862.
The complement is counted by A362610, ranks A359178.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A053263, A098859, A304442, A353864, A360071, A362612.

Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]>1&]],{n,0,30}]

A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 100, 102, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 128, 130, 132, 134, 136, 138

Offset: 1

Amiram Eldar, Jan 21 2023

Numbers k such that A007814(k) = A051903(k).
The powers of 2 (A000079) are all terms.
The product of any two terms (not necessarily distinct) is also a term.
This sequence is a disjoint union of {1} and the subsequences of numbers m of the form 2^(k-1)*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 2*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 4*o where o is an odd cubefree number; etc.
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 0.44541445377638761933... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} (k-1)/(zeta(k)*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 2.10346728882748723133... . [corrected by Amiram Eldar, Jul 10 2025]
Also numbers whose multiset of prime factors has low (i.e. least) mode 2. Here, 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}. - Gus Wiseman, Jul 14 2023

			From _Gus Wiseman_, Jul 14 2023: (Start)
108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is present because it has low mode 2.
The terms together with their prime factorizations begin:
   1 =
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  10 = 2*5
  12 = 2*2*3
  14 = 2*7
  16 = 2*2*2*2
  20 = 2*2*5
  22 = 2*11
  24 = 2*2*2*3
  26 = 2*13
  28 = 2*2*7
  30 = 2*3*5
  32 = 2*2*2*2*2
  34 = 2*17
  36 = 2*2*3*3
(End)

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

Cf. A000079, A000265, A007814, A051903, A056911.
Partitions of this type are counted by A241131.
The case of unique mode is A360013, complement here A360014.
For unique minimal prime exponent we have A364061, counted by A364062.
For minimal prime exponent we have A364158, counted by A364159.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
Cf. A002865, A327473, A327476, A356862, A359178, A362605, A362613, A363486.

q[n_] := IntegerExponent[n, 2] == Max[FactorInteger[n][[;; , 2]]]; q[1] = True; Select[Range[150], q]

is(n) = n == 1 || vecmax(factor(n)[,2]) == valuation(n, 2);

Disjoint union of A360013 and A360014.

a(n) = A360013(n)/2. - Gus Wiseman, Jul 14 2023

A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A130092 in lacking 180.
First differs from A351295 in lacking 180 and having 216.
First differs from A362605 in having 60.

			The prime factorization of 1800 is {2,2,2,3,3,5,5}, and the parts of least multiplicity are {3,5}, so 1800 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}

The complement is A359178, counted by A362610.
For mode we have A362605, counted by A362607.
Partitions of this type are counted by A362609.
These are the positions of terms > 1 in A362613.
A112798 lists prime indices, length A001222, sum A056239.
A362614 counts partitions by number of modes.
A362615 counts partitions by number of co-modes.
Cf. A215366, A327473, A327476, A353864, A356862, A359908, A362611.

Select[Range[100],Count[Last/@FactorInteger[#],Min@@Last/@FactorInteger[#]]>1&]

A363486 Low mode in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 23 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}.
Extending the terminology of A124943, the "low mode" in a multiset is its least mode.

Positions of first appearances are 1 and A000040.
Positions of 1's are A360013, counted by A241131.
For greatest instead of least we have A363487.
The version for median is A363941, triangle A124943.
The high version for median is A363942, triangle A124944.
The version for mean instead of mode is A363943, high A363944.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A360015, A363723.

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]&];
Table[If[n==1,0,First[modes[prix[n]]]],{n,30}]

A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Jun 23 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:
     2: {1}          29: {10}              79: {22}
     3: {2}          31: {11}              81: {2,2,2,2}
     4: {1,1}        32: {1,1,1,1,1}       83: {23}
     5: {3}          37: {12}              89: {24}
     7: {4}          41: {13}              90: {1,2,2,3}
     8: {1,1,1}      43: {14}              97: {25}
     9: {2,2}        47: {15}             101: {26}
    11: {5}          49: {4,4}            103: {27}
    13: {6}          53: {16}             107: {28}
    16: {1,1,1,1}    59: {17}             109: {29}
    17: {7}          61: {18}             113: {30}
    19: {8}          64: {1,1,1,1,1,1}    121: {5,5}
    23: {9}          67: {19}             125: {3,3,3}
    25: {3,3}        71: {20}             127: {31}
    27: {2,2,2}      73: {21}             128: {1,1,1,1,1,1,1}

These partitions are counted by A363719, factorizations A363741.
For unequal instead of equal we have A363730, counted by A363720.
Excluding primes gives A363722.
Excluding prime-powers gives A363729, counted by A363728.
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. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725.

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],{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

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

A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 104, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 184, 188, 192, 200, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256

Offset: 1

Amiram Eldar, Jan 21 2023

Numbers k such that A007814(k) > A051903(A000265(k)).
The powers of 2 (A000079), except for 1, are all terms.
The product of any two terms (not necessarily distinct) is also a term.
This sequence is a disjoint union of {2} and the subsequences of numbers m of the form 2^k*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 4*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 8*o where o is an odd cubefree number; etc.
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 0.222707226888193809... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} k/(zeta(k)*2*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 3.10346728882748723133... . [corrected by Amiram Eldar, Jul 10 2025]
This sequence is a subsequence of A360015 and the asymptotic density of this sequence within A360015 is exactly 1/2.
Also even numbers whose multiset of prime factors has unique mode 2. - Gus Wiseman, Jul 10 2023

			From _Gus Wiseman_, Jul 09 2023: (Start)
108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is missing because 2 is not the unique mode.
120 = 2*2*2*3*5 is present because its unique mode is 2.
The terms together with their prime factorizations begin:
   2 = 2
   4 = 2*2
   8 = 2*2*2
  12 = 2*2*3
  16 = 2*2*2*2
  20 = 2*2*5
  24 = 2*2*2*3
  28 = 2*2*7
  32 = 2*2*2*2*2
  40 = 2*2*2*5
  44 = 2*2*11
  48 = 2*2*2*2*3
  52 = 2*2*13
  56 = 2*2*2*7
  60 = 2*2*3*5
  64 = 2*2*2*2*2*2
(End)

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

Cf. A000079, A000265, A007814, A051903, A056911.
Equals A360015 \ A360014.
Partitions of this type are counted by A241131.
Allowing any unique mode gives A356862, complement A362605.
Allowing any unique co-mode gives A359178, complement A362606.
Not requiring the mode to be unique gives A360015.
The opposite version is A362616, counted by A362612.
For co-mode instead of mode we have A364061, counted by A364062.
With least prime factor instead of 2, we have A364160, counted by A364193.
With a different factorization, we have the subsequence A335738.
A124010 gives prime signature, ordered A118914.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Cf. A001222, A002865, A327473, A327476, A362608, A362610, A363723, A363727.

q[n_] := Module[{e = IntegerExponent[n, 2], m}, m = n/2^e; (m == 1 && e > 0) || AllTrue[FactorInteger[m][[;; , 2]], # < e &]]; Select[Range[256], q]

is(n) = {my(e = valuation(n, 2), m = n >> e); (m == 1 && e > 0) || (m > 1 && vecmax(factor(m)[,2]) < e)};

a(n) = 2*A360015(n). - Gus Wiseman, Jul 10 2023

A363487 High mode in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 04 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}.
Extending the terminology of A124944, the "high mode" in a multiset is its greatest mode.

Positions of first appearances are 1 and A000040.
Positions of 1's are A360015, counted by A241131.
For low instead of high mode we have A363486.
The version for low median is A363941, triangle A124943.
The version for high median is A363942, triangle A124944.
The version for mean instead of mode is A363944, low A363943.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A359908, A360013, A363723, A363729.

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]&];
Table[If[n==1,0,Last[modes[prix[n]]]],{n,30}]

A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 150, 151, 157, 162, 163, 167

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A329131 in lacking 450 and having 1500.

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

			The factorization of 90 is 2*3*3*5, modes {3}, so 90 is missing.
The factorization of 450 is 2*3*3*5*5, modes {3,5}, so 450 is missing.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so 900 is missing.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so 1500 is present.
The terms together with their prime indices begin:
     2: {1}          27: {2,2,2}           67: {19}
     3: {2}          29: {10}              71: {20}
     4: {1,1}        31: {11}              73: {21}
     5: {3}          32: {1,1,1,1,1}       75: {2,3,3}
     7: {4}          37: {12}              79: {22}
     8: {1,1,1}      41: {13}              81: {2,2,2,2}
     9: {2,2}        43: {14}              83: {23}
    11: {5}          47: {15}              89: {24}
    13: {6}          49: {4,4}             97: {25}
    16: {1,1,1,1}    50: {1,3,3}           98: {1,4,4}
    17: {7}          53: {16}             101: {26}
    18: {1,2,2}      54: {1,2,2,2}        103: {27}
    19: {8}          59: {17}             107: {28}
    23: {9}          61: {18}             108: {1,1,2,2,2}
    25: {3,3}        64: {1,1,1,1,1,1}    109: {29}

First term with given bigomega is A000079.
For median instead of mode we have A053263.
Partitions of this type are counted by A362612.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362614 counts partitions by number of modes, ranked by A362611.
A362615 counts partitions by number of co-modes, ranked by A362613.
Cf. A000040, A002865, A327473, A327476, A358137, A360687.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Commonest[prifacs[#]]=={Max[prifacs[#]]}&]

A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 46, 61, 82, 102, 136, 174, 224, 283, 360, 449, 569, 708, 883, 1089, 1352, 1659, 2042, 2492, 3039, 3695, 4492, 5426, 6555, 7889, 9482, 11360, 13602, 16231, 19348, 23005, 27313, 32364, 38303, 45227, 53341, 62800, 73829

Offset: 1

Gus Wiseman, Jun 26 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

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) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (3221)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For mean instead of mode we have A240219, see A359894, A359889, A359895, A359897, A359899.
Including mean also gives A363719, ranks A363727.
For mean instead of median we have A363723, see A363724, A363731.
A000041 counts integer 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, ranks A356862.
Cf. A027193, A237984, A325347, A362562, A363720, A363725, A363726.

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

A363731 Number of integer partitions of n whose mean is a mode but not the only mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 5, 0, 1, 8, 5, 0, 12, 0, 19, 14, 2, 0, 52, 21, 3, 23, 59, 0, 122, 0, 97, 46, 6, 167, 303, 0, 8, 82, 559, 0, 543, 0, 355, 745, 15, 0, 1685, 510, 1083, 251, 840, 0, 2325, 1832, 3692, 426, 34, 0, 9599

Offset: 0

Gus Wiseman, Jun 24 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(n) partitions for n = 6, 9, 12, 15, 18:
  (3,2,1)  (4,3,2)  (5,4,3)        (6,5,4)      (7,6,5)
           (5,3,1)  (6,4,2)        (7,5,3)      (8,6,4)
                    (7,4,1)        (8,5,2)      (9,6,3)
                    (6,3,2,1)      (9,5,1)      (10,6,2)
                    (3,3,2,2,1,1)  (4,4,3,3,1)  (11,6,1)
                                   (5,3,3,2,2)  (4,4,3,3,2,2)
                                   (5,4,3,2,1)  (5,5,3,3,1,1)
                                   (7,3,3,1,1)  (6,4,3,3,1,1)
                                                (7,3,3,2,2,1)
                                                (8,3,3,2,1,1)
                                                (3,3,3,2,2,2,1,1,1)
                                                (6,2,2,2,2,1,1,1,1)

For a unique mode we have A363723, non-constant A362562.
For any number of modes we have A363724.
A000041 counts 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 with a unique mode, ranks A356862.
A363719 counts partitions with all three averages equal, ranks A363727.
Cf. A240219, A326567/A326568, A359893, A363720, A363725, A363740.

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

cp's OEIS Frontend

A362609 Number of integer partitions of n with more than one part of least multiplicity.

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A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.

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A362606 Numbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

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A363486 Low mode in the multiset of prime indices of n.

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A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.

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A363487 High mode in the multiset of prime indices of n.

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A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

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A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

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