A241131 -id:A241131 - cp's OEIS frontend

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

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

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}]

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}]

A363949 Numbers whose prime indices have mean 1 when rounded down.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 560, 576, 600, 640

Offset: 1

Gus Wiseman, Jul 02 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}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

These partitions are counted by A025065.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A360015, counted by A241131.
For median instead of mean we have A363488, counted by A027336.
Positions of 1's in A363943, triangle A363945.
For the usual rounding (not low or high) we have A363948, counted by A363947.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
For mean 2 instead of 1 we have A363950, counted by A026905 redoubled.
Cf. A051293, A327473, A327476, A344296, A359889, A360013, A363723, A363727, A363946, A363951.

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

a(n) = 2*A344296(n).

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

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 3, 1, 0, 0, 3, 4, 1, 2, 0, 0, 4, 7, 2, 1, 0, 0, 0, 5, 9, 4, 1, 2, 0, 0, 0, 6, 13, 4, 4, 1, 0, 0, 0, 0, 8, 18, 6, 3, 2, 3, 0, 0, 0, 0, 10, 26, 9, 5, 2, 2, 0, 0, 0, 0, 0, 12, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset: 1

Gus Wiseman, Mar 01 2025

The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Triangle begins:
   1
   1  1
   1  0  2
   2  1  0  2
   3  1  0  0  3
   4  1  2  0  0  4
   7  2  1  0  0  0  5
   9  4  1  2  0  0  0  6
  13  4  4  1  0  0  0  0  8
  18  6  3  2  3  0  0  0  0 10
  26  9  5  2  2  0  0  0  0  0 12
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Last column (k=n) is A000009.
Row sums are A000041.
Row sums without the last column (k=n) are A047967.
For first instead of last part we have A116861, rank A066328.
First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
Using Heinz numbers, this statistic is given by A381437.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
Look-and-Say partition: A048767, A351294, A351295, A381440.
Cf. A047966, A051903, A051904, A091602, A181819, A212166.

egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]

A363947 Number of integer partitions of n with mean < 3/2.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 7, 12, 12, 12, 19, 19, 19, 30, 30, 30, 45, 45, 45, 67, 67, 67, 97, 97, 97, 139, 139, 139, 195, 195, 195, 272, 272, 272, 373, 373, 373, 508, 508, 508, 684, 684, 684, 915, 915, 915, 1212, 1212, 1212, 1597, 1597, 1597, 2087

Offset: 0

Gus Wiseman, Jul 02 2023

nonn

			The partition y = (2,2,1) has mean 5/3, which is not less than 3/2, so y is not counted under 5.
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (11)  (111)  (211)   (2111)   (21111)   (22111)    (221111)
                    (1111)  (11111)  (111111)  (31111)    (311111)
                                               (211111)   (2111111)
                                               (1111111)  (11111111)

The high version is A000012 (all ones).
This is A000070 with each term repeated three times (see A025065 for two).
These partitions have ranks A363948.
The complement is counted by A364059.
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.
A327482 counts partitions by integer mean.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A026905, A027336, A237984, A241131, A327472, A363724, A363745, A363943, A363949.

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

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}]

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

Original entry on oeis.org

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}]

cp's OEIS Frontend

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

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

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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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A363949 Numbers whose prime indices have mean 1 when rounded down.

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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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Maple

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A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

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A363947 Number of integer partitions of n with mean < 3/2.

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