A363723 -id:A363723 - cp's OEIS frontend

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

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}]

A363946 Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 3, 2, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 6, 4, 3, 0, 0, 1, 0, 1, 11, 5, 4, 0, 0, 0, 1, 0, 1, 11, 13, 0, 4, 0, 0, 0, 1, 0, 1, 18, 9, 8, 5, 0, 0, 0, 0, 1, 0, 1, 18, 21, 10, 0, 5, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jun 30 2023

Extending the terminology of A124944, the "high mean" of a multiset is obtained by taking the mean and rounding up.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  3  2  0  1
  0  1  6  3  0  0  1
  0  1  6  4  3  0  0  1
  0  1 11  5  4  0  0  0  1
  0  1 11 13  0  4  0  0  0  1
  0  1 18  9  8  5  0  0  0  0  1
  0  1 18 21 10  0  5  0  0  0  0  1
  0  1 29 28 12  0  6  0  0  0  0  0  1
  0  1 29 32 18 14  0  6  0  0  0  0  0  1
  0  1 44 43 23 16  0  7  0  0  0  0  0  0  1
  0  1 44 77 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (1111111)  (4111)    (511)  (61)  .  .  (7)
                (3211)    (421)  (52)
                (31111)   (331)  (43)
                (2221)    (322)
                (22111)
                (211111)

Row sums are A000041.
Column k = 2 is A026905 redoubled, ranks A363950.
For median instead of mean we have triangle A124944, low A124943.
For mode instead of mean we have rank stat A363486, high A363487.
For median instead of mean we have rank statistic A363942, low A363941.
The rank statistic for this triangle is A363944.
The version for low mean is A363945, rank statistic A363943.
For mode instead of mean we have triangle A363953, low A363952.
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. A002865, A025065, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363948.

meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meanup[#]==k&]],{n,0,15},{k,0,n}]

A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 4, 2, 0, 0, 1, 0, 4, 3, 3, 0, 0, 1, 0, 7, 4, 3, 0, 0, 0, 1, 0, 7, 10, 0, 4, 0, 0, 0, 1, 0, 12, 6, 7, 4, 0, 0, 0, 0, 1, 0, 12, 16, 8, 0, 5, 0, 0, 0, 0, 1, 0, 19, 21, 10, 0, 5, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 30 2023

Extending the terminology of A124943, the "low mean" of a multiset is its mean rounded down.

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  4  2  0  0  1
  0  4  3  3  0  0  1
  0  7  4  3  0  0  0  1
  0  7 10  0  4  0  0  0  1
  0 12  6  7  4  0  0  0  0  1
  0 12 16  8  0  5  0  0  0  0  1
  0 19 21 10  0  5  0  0  0  0  0  1
  0 19 24 15 12  0  6  0  0  0  0  0  1
  0 30 32 18 14  0  6  0  0  0  0  0  0  1
  0 30 58 23 16  0  0  7  0  0  0  0  0  0  1
  0 45 47 57  0 19  0  7  0  0  0  0  0  0  0  1
Row k = 8 counts the following partitions:
  .  (41111)     (611)   .  (71)  .  .  .  (8)
     (32111)     (521)      (62)
     (311111)    (5111)     (53)
     (22211)     (431)      (44)
     (221111)    (422)
     (2111111)   (4211)
     (11111111)  (332)
                 (3311)
                 (3221)
                 (2222)

Row sums are A000041.
Column k = 1 is A025065, ranks A363949.
For median instead of mean we have triangle A124943, high A124944.
Column k = 2 is A363745.
For median instead of mean we have rank statistic A363941, high A363942.
The rank statistic for this triangle is A363943.
The high version is A363946, rank statistic A363944.
For mode instead of mean we have A363952, rank statistic A363486.
For high mode instead of mean we have A363953, rank statistic A363487.
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. A002865, A026905, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363951.

meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meandown[#]==k&]],{n,0,15},{k,0,n}]

A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 3, 8, 8, 17, 19, 28, 39, 59, 68, 106, 123, 165, 220, 301, 361, 477, 605, 745, 929, 1245, 1456, 1932, 2328, 2846, 3590, 4292, 5111, 6665, 8040, 9607, 11532, 14410, 16699, 20894, 24287, 28706, 35745, 42845, 49548, 59963, 70985

Offset: 0

Gus Wiseman, Jun 22 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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(7) = 1 through a(13) = 17 partitions:
  (3211)  (4211)  (3321)  (5311)    (4322)    (4431)    (4432)
                  (4311)  (6211)    (4421)    (5322)    (5422)
                  (5211)  (322111)  (5411)    (6411)    (5521)
                                    (6311)    (7311)    (6322)
                                    (7211)    (8211)    (6511)
                                    (43211)   (53211)   (7411)
                                    (332111)  (432111)  (8311)
                                    (422111)  (522111)  (9211)
                                                        (54211)
                                                        (63211)
                                                        (333211)
                                                        (433111)
                                                        (442111)
                                                        (532111)
                                                        (622111)
                                                        (3322111)
                                                        (32221111)

The length-4 case appears to be A325695.
For equal instead of unequal we have A363719, ranks A363727.
Allowing multiple modes gives A363720, ranks A363730.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A359896, A359900, A363723, A363728.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]],{n,0,30}]

A363730 Numbers whose prime indices have different mean, median, and mode.

Original entry on oeis.org

42, 60, 66, 70, 78, 84, 102, 114, 130, 132, 138, 140, 150, 154, 156, 165, 170, 174, 180, 182, 186, 190, 195, 204, 220, 222, 228, 230, 231, 246, 255, 258, 260, 266, 276, 282, 285, 286, 290, 294, 308, 310, 315, 318, 322, 330, 340, 345, 348, 354, 357, 360, 364

Offset: 1

Gus Wiseman, Jun 24 2023

nonn

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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 prime indices of 180 are {1,1,2,2,3}, with mean 9/5, median 2, modes {1,2}, so 180 is in the sequence.
The prime indices of 108 are {1,1,2,2,2}, with mean 8/5, median 2, modes {2}, so 108 is not in the sequence.
The terms together with their prime indices begin:
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}
  140: {1,1,3,4}
  150: {1,2,3,3}

These partitions are counted by A363720
For equal instead of unequal we have A363727, counted by A363719.
The version for factorizations is A363742, equal A363741.
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, A359908, A363722, A363725, 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]&];
Select[Range[100],{Mean[prix[#]]}!={Median[prix[#]]}!=modes[prix[#]]&]

All three of A326567(a(n))/A326568(a(n)), A360005(a(n))/2, and A363486(a(n)) = A363487(a(n)) are different.

A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888

Offset: 1

Gus Wiseman, Jul 12 2023

nonn

Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is 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 terms together with their prime factors begin:
    2 = 2
    4 = 2*2
    8 = 2*2*2
   16 = 2*2*2*2
   18 = 2*3*3
   32 = 2*2*2*2*2
   50 = 2*5*5
   54 = 2*3*3*3
   64 = 2*2*2*2*2*2
   98 = 2*7*7
  108 = 2*2*3*3*3
  128 = 2*2*2*2*2*2*2

Robert Israel, Table of n, a(n) for n = 1..10000

For any unique co-mode: A359178, counted by A362610, complement A362606.
For high mode: A360013, positions of 1's in A363487, counted by A241131.
For low mode: A360015, positions of 1's in A363486, counted by A241131.
Partitions of this type are counted by A364062.
For low co-mode: A364158, positions of 1's in A364192, counted by A364159.
Positions of 1's in A364191, high A364192.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions w/ unique mode, count A362608, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.

filter:= proc(n) local F,F2,Fo;
  F:= ifactors(n)[2];
  F2,Fo:= selectremove(t -> t[1]=2, F);
  Fo = [] or F2[1,2] < min(Fo[..,2])
end proc:
select(filter, 2*[$1..5000]); # Robert Israel, Apr 22 2024

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Select[Range[100],comodes[prifacs[#]]=={2}&]

from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(),default=0) or m==1, count(max(startvalue+startvalue&1,2),2))
A364061_list = list(islice(A364061_gen(),30)) # Chai Wah Wu, Jul 14 2023

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} (1-1/2^(k-1))*(s(k)-s(k+1)) = 1.16896822653093929144..., where s(k) = Product_{primes p >= 3} (1 + 1/(p^(k-1)*(p-1))) is the sum of reciprocals of the odd k-full numbers (numbers whose prime factorization has no exponent that is smaller than k). - Amiram Eldar, Aug 30 2024

A363952 Number of integer partitions of n with low mode k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 4, 2, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 9, 3, 2, 0, 0, 0, 1, 0, 13, 5, 2, 1, 0, 0, 0, 1, 0, 18, 6, 3, 2, 0, 0, 0, 0, 1, 0, 26, 9, 3, 2, 1, 0, 0, 0, 0, 1, 0, 32, 13, 5, 3, 2, 0, 0, 0, 0, 0, 1, 0, 47, 16, 7, 3, 2, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jul 07 2023

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" of a multiset is the least mode.

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   0   1
   0   3   1   0   1
   0   4   2   0   0   1
   0   7   2   1   0   0   1
   0   9   3   2   0   0   0   1
   0  13   5   2   1   0   0   0   1
   0  18   6   3   2   0   0   0   0   1
   0  26   9   3   2   1   0   0   0   0   1
   0  32  13   5   3   2   0   0   0   0   0   1
   0  47  16   7   3   2   1   0   0   0   0   0   1
   0  60  21  10   4   3   2   0   0   0   0   0   0   1
   0  79  30  13   6   3   2   1   0   0   0   0   0   0   1
   0 104  38  17   7   4   3   2   0   0   0   0   0   0   0   1
Row n = 8 counts the following partitions:
  .  (71)        (62)     (53)   (44)  .  .  .  (8)
     (611)       (422)    (332)
     (521)       (3221)
     (5111)      (2222)
     (431)       (22211)
     (4211)
     (41111)
     (3311)
     (32111)
     (311111)
     (221111)
     (2111111)
     (11111111)

Row sums are A000041.
For median: A124943 (high A124944), rank statistic A363941 (high A363942).
Column k = 1 is A241131 (partitions w/ low mode 1), ranks A360015, A360013.
The rank statistic for this triangle is A363486.
For mean: A363945 (high A363946), rank statistic A363943 (high A363944).
The high version is A363953.
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, A067538, A237984, A363723, A363724, A363731.

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

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

90, 270, 525, 550, 756, 810, 1666, 1911, 1950, 2268, 2430, 2625, 2695, 2700, 2750, 5566, 6762, 6804, 6897, 7128, 7290, 8100, 8500, 9310, 9750, 10285, 10478, 11011, 11550, 11662, 12250, 12375, 12495, 13125, 13377, 13750, 14014, 14703, 18865, 19435, 20412, 21384

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 prime indices of 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
The terms together with their prime indices begin:
     90: {1,2,2,3}
    270: {1,2,2,2,3}
    525: {2,3,3,4}
    550: {1,3,3,5}
    756: {1,1,2,2,2,4}
    810: {1,2,2,2,2,3}
   1666: {1,4,4,7}
   1911: {2,4,4,6}
   1950: {1,2,3,3,6}
   2268: {1,1,2,2,2,2,4}
   2430: {1,2,2,2,2,2,3}

For just primes instead of prime powers we have A363722.
Including prime-powers gives A363727, counted by A363719.
These partitions are counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A000961 lists the prime powers, complement A024619.
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, 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[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

A363741 Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 26 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
Position of first appearance of n is: (1, 2, 4, 16, 64, 5832, 4096, ...).

			The factorization 6*9*9*12 = 5832 has mean 9, median 9, and modes {9}, so it is counted under a(5832).
The a(n) factorizations for selected n:
2   4     16        64            5832              4096
    2*2   4*4       8*8           18*18*18          64*64
          2*2*2*2   4*4*4         6*9*9*12          8*8*8*8
                    2*2*2*2*2*2   3*6*6*6*9         16*16*16
                                  2*3*3*3*3*3*3*4   4*4*4*4*4*4
                                                    2*2*2*2*2*2*2*2*2*2*2*2

For just (mean) = (median): A359909, see A240219, A359889, A359910, A359911.
The version for partitions is A363719, unequal A363720.
For unequal instead of equal we have A363742.
A000041 counts integer partitions.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A326622 counts factorizations with integer mean, strict A328966.
A339846 counts even-length factorizations, A339890 odd-length.
Cf. A237984, A326567/A326568, A327472, A359893, A363723, A363727, A363740.

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],{Mean[#]}=={Median[#]}==modes[#]&]],{n,100}]

A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

Original entry on oeis.org

2, 9, 10, 68, 78, 98, 99, 105, 110, 125, 328, 444, 558, 620, 783, 812, 870, 966, 988, 1012, 1035, 1150, 1156, 1168, 1197, 1254, 1326, 1330, 1425, 1521, 1666, 1683, 1690, 1704, 1785, 1870, 1911, 2002, 2125, 2145, 2275, 2401, 2412, 2541, 2662, 2680, 2695, 3025

Offset: 1

Gus Wiseman, Jul 05 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.

			The terms together with their prime indices begin:
    2: {1}
    9: {2,2}
   10: {1,3}
   68: {1,1,7}
   78: {1,2,6}
   98: {1,4,4}
   99: {2,2,5}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  328: {1,1,1,13}
  444: {1,1,2,12}
  558: {1,2,2,11}
  620: {1,1,3,11}
  783: {2,2,2,10}
  812: {1,1,4,10}
  870: {1,2,3,10}
  966: {1,2,4,9}
  988: {1,1,6,8}

Partitions of this type are counted by A364055, without  zeros A206240.
The RHS is A001222.
The LHS is A326567/A326568.
A008284 counts partitions by length, A058398 by mean.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, sum A056239.
A124943 counts partitions by low median, high A124944.
A316413 ranks partitions with integer mean, counted by A067538.
A326622 counts factorizations with integer mean, strict A328966.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A025065, A327473, A327476, A327482, A359889, A363723, A363727, A363729, A363944, A363949.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]==PrimeOmega[#]&]

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A363946 Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k.

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A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.

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A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

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A363730 Numbers whose prime indices have different mean, median, and mode.

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A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

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A363952 Number of integer partitions of n with low mode k.

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A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A363741 Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

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