A118096 -id:A118096 - cp's OEIS frontend

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954

Offset: 1

Clark Kimberling, Feb 16 2014

nonn

By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - Gus Wiseman, May 14 2023

			a(6) = 7 counts these partitions:  6, 42, 33, 222, 2211, 21111, 111111.
From _Gus Wiseman_, May 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (322)      (53)
                    (1111)  (2111)   (222)     (2221)     (332)
                            (11111)  (2211)    (22111)    (422)
                                     (21111)   (211111)   (2222)
                                     (111111)  (1111111)  (22211)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (2211)    (2221)     (332)
                                     (111111)  (1111111)  (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..300 from John Tyler Rascoe)

Cf. A237821, A118096, A053263.
The complement is counted by A237820, ranks A362982.
For modes instead of middles we have A362619, counted by A171979.
These partitions have ranks A362981.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622, A384426.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 13 2025 *)
(* or *)
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^k/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 14 2025 *)

N=60; x='x+O('x^N);
gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
Vec(gf) \\ John Tyler Rascoe, Mar 07 2024

G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - John Tyler Rascoe, Mar 07 2024
G.f.: Sum_{k>=1} x^k / Product_{j=k..2*k} (1 - x^j). - Vaclav Kotesovec, Jun 13 2025
a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4) * sqrt(2*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 14 2025

A361849 Number of integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 3, 4, 7, 9, 9, 15, 16, 20, 26, 34, 37, 50, 55, 68, 86, 103, 117, 145, 168, 201, 236, 282, 324, 391, 449, 525, 612, 712, 818, 962, 1106, 1278, 1470, 1698, 1939, 2238, 2550, 2924, 3343, 3824, 4341, 4963, 5627, 6399, 7256, 8231, 9300

Offset: 1

Gus Wiseman, Apr 02 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).

			The a(4) = 1 through a(11) = 9 partitions:
  211  2111  21111  421     422      4221      631        632
                    3211    221111   4311      4222       5321
                    22111   2111111  2211111   42211      5411
                    211111           21111111  322111     42221
                                               2221111    43211
                                               22111111   332111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
For example, the partition (3,2,1,1) has maximum 3 and median 3/2, so is counted under a(7).

For minimum instead of median we have A118096.
For length instead of median we have A237753.
This is the equal case of A361848.
For mean instead of median we have A361853.
These partitions have ranks A361856.
For "greater" instead of "equal" we have A361857, allowing equality A361859.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361860 counts partitions with minimum equal to median.
Cf. A013580, A027193, A053263, A061395, A067659, A111907, A116608, A237755, A240219, A361851.

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

A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

Original entry on oeis.org

12, 24, 42, 48, 60, 63, 72, 96, 126, 130, 140, 144, 189, 192, 195, 252, 266, 288, 308, 325, 330, 360, 378, 384, 399, 420, 432, 495, 546, 567, 572, 576, 588, 600, 630, 638, 650, 665, 756, 768, 819, 864, 882, 884, 931, 945, 957, 962, 975, 1122, 1134, 1152, 1190

Offset: 1

Gus Wiseman, Apr 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
These are Heinz numbers of partitions satisfying (maximum) = 2*(median).

			The terms together with their prime indices begin:
    12: {1,1,2}
    24: {1,1,1,2}
    42: {1,2,4}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
    63: {2,2,4}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   126: {1,2,2,4}
   130: {1,3,6}
   140: {1,1,3,4}
   144: {1,1,1,1,2,2}
The prime indices of 126 are {1,2,2,4}, with maximum 4 and median 2, so 126 is in the sequence.
The prime indices of 308 are {1,1,4,5}, with maximum 5 and median 5/2, so 308 is in the sequence.

The LHS (greatest prime index) is A061395.
The RHS (twice median) is A360005, distinct A360457.
These partitions are counted by A361849.
For mean instead of median we have A361855, counted by A361853.
For minimum instead of median we have A361908, counted by A118096.
For length instead of median we have A361909, counted by A237753.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A067801, A111907, A361205, A361848, A361858.

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

A061395(a(n)) = 2*A360005(a(n)).

A237821 Number of partitions of n such that 2*(least part) <= greatest part.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 16, 25, 35, 48, 68, 92, 123, 164, 216, 282, 367, 471, 604, 769, 975, 1225, 1542, 1924, 2395, 2968, 3669, 4514, 5547, 6781, 8280, 10071, 12229, 14796, 17881, 21537, 25902, 31066, 37206, 44443, 53021, 63098, 74995, 88946, 105350, 124533

Offset: 1

Clark Kimberling, Feb 16 2014

By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023

			a(6) = 7 counts these partitions:  51, 42, 411, 321, 3111, 2211, 21111.
From _Gus Wiseman_, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
  (21)  (31)   (41)    (42)     (52)
        (211)  (221)   (51)     (61)
               (311)   (321)    (331)
               (2111)  (411)    (421)
                       (2211)   (511)
                       (3111)   (2221)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
  (21)  (31)   (32)    (42)     (43)
        (211)  (41)    (51)     (52)
               (311)   (321)    (61)
               (2111)  (411)    (322)
                       (2211)   (421)
                       (3111)   (511)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
(End)

The complement is counted by A053263, ranks A081306.
These partitions have ranks A069900.
The case of equality is A118096.
For < instead of <= we have A237820, ranks A362982.
For >= instead of <= we have A237824, ranks A362981.
The conjugate partitions have ranks A362980.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023

A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also partitions satisfying (maximum) = 2*(mean).

These are partitions whose diagram has the same size as its complement (see example).

			The a(6) = 2 through a(12) = 10 partitions:
  (411)   .  (4211)  (621)     (5221)   .  (822)
  (3111)             (321111)  (5311)      (831)
                               (42211)     (6222)
                               (43111)     (6321)
                                           (6411)
                                           (422211)
                                           (432111)
                                           (441111)
                                           (32211111)
                                           (33111111)
The partition y = (6,4,1,1) has diagram:
  o o o o o o
  o o o o . .
  o . . . . .
  o . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).

For minimum instead of mean we have A118096.
For length instead of mean we have A237753.
For median instead of mean we have A361849, ranks A361856.
This is the equal case of A361851, unequal case A361852.
The strict case is A361854.
These partitions have ranks A361855.
This is the equal case of A361906, unequal case A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

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

A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

Original entry on oeis.org

28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254

Offset: 1

Gus Wiseman, Mar 29 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.
Also positive integers whose prime indices satisfy (maximum) = 2*(mean).
Also Heinz numbers of partitions of the same size as their complement (see example).

			The terms together with their prime indices begin:
   28: {1,1,4}
   40: {1,1,1,3}
   78: {1,2,6}
   84: {1,1,2,4}
  171: {2,2,8}
  190: {1,3,8}
  198: {1,2,2,5}
  220: {1,1,3,5}
  240: {1,1,1,1,2,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
  o o o o o
  o o . . .
  o o . . .
  o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

These partitions are counted by A361853, strict A361854.
For median instead of mean we have A361856, counted by A361849.
For minimum instead of mean we have A361908, counted by A118096.
For length instead of mean we have A361909, counted by A237753.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

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

A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

A361860 Number of integer partitions of n whose median part is the smallest.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 12, 15, 21, 25, 36, 44, 58, 72, 95, 117, 150, 185, 235, 289, 362, 441, 550, 670, 824, 1000, 1223, 1476, 1795, 2159, 2609, 3126, 3758, 4485, 5369, 6388, 7609, 9021, 10709, 12654, 14966, 17632, 20782, 24414, 28684, 33601, 39364, 45996

Offset: 1

Gus Wiseman, Apr 02 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).

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For mean instead of median we have A000005.
For length instead of median we have A006141.
For maximum instead of median we have A053263.
For half-median we have A361861.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A053263, A067659, A111907, A116608, A118096, A237753, A240219, A359907, A361848, A361849.

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

A237820 Number of partitions of n such that 2*(least part) < greatest part.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 8, 12, 19, 29, 42, 58, 83, 112, 151, 202, 267, 347, 453, 581, 744, 948, 1198, 1505, 1889, 2356, 2925, 3621, 4465, 5486, 6724, 8212, 9999, 12151, 14715, 17784, 21442, 25795, 30952, 37079, 44315, 52871, 62950, 74827, 88767, 105159, 124335

Offset: 1

Clark Kimberling, Feb 16 2014

			a(6) = 4 counts these partitions:  51, 411, 321, 3111.

John Tyler Rascoe, Table of n, a(n) for n = 1..200

Cf. A000041, A053263, A118096, A237821, A237824.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

A(n) = {concat([0,0,0], Vec(sum(i=1, n, sum(j=1, n-3*i, x^(3*i+j)/prod(k=i, min(n-3*i-j,2*i+j), 1-x^k)))+ O('x^(n+1))))} \\ John Tyler Rascoe, Jun 21 2025

G.f.: Sum_{i>0} Sum_{j>0} x^(3*i+j) /Product_{k=i..2*i+j} (1 - x^k). - John Tyler Rascoe, Jun 21 2025

A237828 Number of partitions of n such that 2*(least part) + 1 = greatest part.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 10, 12, 17, 18, 22, 27, 31, 34, 42, 45, 53, 61, 66, 72, 86, 92, 103, 113, 125, 135, 154, 163, 180, 197, 213, 229, 257, 271, 294, 318, 346, 368, 404, 426, 463, 497, 532, 564, 616, 651, 700, 747, 798, 844, 912, 962, 1033, 1097, 1167, 1231, 1327, 1397, 1486, 1576, 1677

Offset: 1

Clark Kimberling, Feb 16 2014

Also, the number of partitions p of n such that if h = max(p), then h is an (h,0)-separator of p; for example, a(10) counts these 9 partitions: 181, 271, 361, 262, 451, 352, 343, 23131, 1212121. - Clark Kimberling, Mar 24 2014

			a(8) = 4 counts these partitions:  3311, 3221, 32111, 311111.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A118096, A237829.

Cf. A049820, A237757, A237825, A237826, A237827, A000041.

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] = = Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] = = Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] = = Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 = = Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 = = Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n],?(2*Min[#]+1==Max[#]&)],{n,60}] (* _Harvey P. Dale, Jun 25 2017 *)
kmax = 65;
Sum[x^(3k+1)/Product[1-x^j, {j, k, 2k+1}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(3*k + 1)/(1 - x^k)/(1 - x^(2*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 18 2025 *)

my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(3*k+1)/prod(j=k, 2*k+1, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

G.f.: Sum_{k>=1} x^(3*k+1)/Product_{j=k..2*k+1} (1-x^j). - Seiichi Manyama, May 17 2023

a(n) ~ sqrt(phi) * exp(Pi*sqrt(2*n/15)) / (sqrt(2)* 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 20 2025

A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).

Original entry on oeis.org

6, 12, 18, 21, 24, 36, 48, 54, 63, 65, 72, 96, 105, 108, 133, 144, 147, 162, 189, 192, 216, 288, 315, 319, 324, 325, 384, 432, 441, 455, 481, 486, 525, 567, 576, 648, 715, 731, 735, 768, 845, 864, 931, 945, 972, 1007, 1029, 1152, 1296, 1323, 1403, 1458, 1463

Offset: 1

Gus Wiseman, Apr 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:
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    21: {2,4}
    24: {1,1,1,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    63: {2,2,4}
    65: {3,6}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}

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

The RHS is 2*A055396 (twice minimum).
The LHS is A061395 (greatest prime index).
Partitions of this type are counted by A118096.
For mean instead of minimum we have A361855, counted by A361853.
For median instead of minimum we have A361856, counted by A361849.
For length instead of minimum we have A361909, counted by A237753.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A237820, A237821, A361858.

filter:= proc(n) local F,b;
  if n::even then b:= padic:-ordp(n,3);
     if b = 0 then return false else return n = 2^padic:-ordp(n,2) * 3^b fi
  fi;
  F:= ifactors(n)[2][..,1];
  nops(F) >= 2 and numtheory:-pi(max(F)) = 2*numtheory:-pi(min(F))
end proc:
select(filter, [$1..2000]); # Robert Israel, Mar 11 2025

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimePi[FactorInteger[#][[1,1]]]&]

cp's OEIS Frontend

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

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A361849 Number of integer partitions of n such that the maximum is twice the median.

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A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

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A237821 Number of partitions of n such that 2*(least part) <= greatest part.

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A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

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A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

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A361860 Number of integer partitions of n whose median part is the smallest.

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A237820 Number of partitions of n such that 2*(least part) < greatest part.

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