A359901 -id:A359901 - cp's OEIS frontend

A359909 Number of integer factorizations of n into factors > 1 with the same mean as median.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 4, 2, 2, 2, 6, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 7, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 6, 4, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 4, 1, 4, 5, 2, 1, 6, 1, 4, 2, 5, 1, 4, 2, 3, 3, 2, 2, 11

Offset: 1

Gus Wiseman, Jan 24 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(n) factorizations for n = 24, 36, 60, 120, 144, 360:
  24      36        60      120       144       360
  3*8     4*9       2*30    2*60      2*72      4*90
  4*6     6*6       3*20    3*40      3*48      5*72
  2*12    2*18      4*15    4*30      4*36      6*60
  2*3*4   3*12      5*12    5*24      6*24      8*45
          2*2*3*3   6*10    6*20      8*18      9*40
                    3*4*5   8*15      9*16      10*36
                            10*12     12*12     12*30
                            4*5*6     2*2*6*6   15*24
                            2*6*10    3*3*4*4   18*20
                            2*3*4*5             2*180
                                                3*120
                                                2*10*18
                                                3*4*5*6

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A240219, complement A359894.
These multisets are ranked by A359889.
The version for strict partitions is A359897.
The odd-length case is A359910.
The complement is counted by A359911.
A001055 counts factorizations.
A058398 counts partitions by mean, see also A008284, A327482.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A316313, A326567/A326568, A359906, A360005.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Mean[#]==Median[#]&]],{n,100}]

median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
A359909(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359909(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

More terms from Antti Karttunen, Jan 20 2025

A360249 Numbers for which the prime indices have the same median as the distinct prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 125, 126, 127, 128, 129, 130

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

First differs from A072774 in having 90.
First differs from A242414 in having 180.
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).

			The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is in the sequence.
The prime indices of 180 are {1,1,2,2,3} with median 2 and distinct prime indices {1,2,3} with median 2, so 180 is in the sequence.

These partitions are counted by A360245.
The complement for mean instead of median is A360246, counted by A360242.
For mean instead of median we have A360247, counted by A360243.
The complement is A360248, counted by A360244.
For multiplicities instead of parts: A360453, counted by A360455.
For multiplicities instead of distinct parts: A360454, counted by A360456.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranks A359889.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A325347 = partitions with integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median.
A359894 = partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
Cf. A000975, A078174, A316413, A324570, A359903, A360252, A360253.

isA360249 := proc(n)
    local ifs,pidx,pe,medAll,medDist ;
    if n = 1 then
        return true ;
    end if ;
    ifs := ifactors(n)[2] ;
    pidx := [] ;
    for pe in ifs do
        numtheory[pi](op(1,pe)) ;
        pidx := [op(pidx),seq(%,i=1..op(2,pe))] ;
    end do:
    medAll := stats[describe,median](sort(pidx)) ;
    pidx := convert(convert(pidx,set),list) ;
    medDist := stats[describe,median](sort(pidx)) ;
    if medAll = medDist then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 130 do
    if isA360249(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 22 2023

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

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103

Offset: 1

Gus Wiseman, Feb 16 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 prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

A359896 Number of odd-length integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 6, 9, 11, 15, 27, 32, 50, 58, 72, 112, 149, 171, 246, 286, 359, 477, 630, 773, 941, 1181, 1418, 1749, 2289, 2668, 3429, 4162, 4878, 6074, 7091, 8590, 10834, 12891, 15180, 18491, 22314, 25845, 31657, 36394, 42269, 52547, 62414, 73576, 85701

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(4) = 1 through a(9) = 11 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (21111)  (331)    (422)      (522)
                         (421)    (431)      (621)
                         (511)    (521)      (711)
                         (22111)  (611)      (22221)
                         (31111)  (22211)    (32211)
                                  (32111)    (33111)
                                  (41111)    (42111)
                                  (2111111)  (51111)
                                             (2211111)
                                             (3111111)

These partitions are ranked by A359892.
The any-length version is A359894, complement A240219, strict A359898.
The complement is counted by A359895, ranked by A359891.
The strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008289, A066571, A316313, A327472, A327482, A359890, A359906, A359910.

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]!=Median[#]&]],{n,0,30}]

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 6, 5, 11, 12, 14, 21, 29, 26, 44, 44, 58, 68, 92, 92, 118, 137, 165, 192, 241, 223, 324, 353, 405, 467, 518, 594, 741, 809, 911, 987, 1239, 1276, 1588, 1741, 1823, 2226, 2566, 2727, 3138, 3413, 3905, 4450, 5093, 5434, 6134

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(7) = 1 through a(13) = 11 partitions:
  (4,2,1)  (4,3,1)  (6,2,1)  (5,3,2)  (5,4,2)    (6,5,1)    (6,4,3)
           (5,2,1)           (5,4,1)  (6,3,2)    (7,3,2)    (6,5,2)
                             (6,3,1)  (6,4,1)    (8,3,1)    (7,4,2)
                             (7,2,1)  (7,3,1)    (9,2,1)    (7,5,1)
                                      (8,2,1)    (6,3,2,1)  (8,3,2)
                                      (5,3,2,1)             (8,4,1)
                                                            (9,3,1)
                                                            (10,2,1)
                                                            (5,4,3,1)
                                                            (6,4,2,1)
                                                            (7,3,2,1)

The non-strict version is ranked by A359890, complement A359889.
The non-strict version is A359894, complement A240219.
The complement is counted by A359897.
The odd-length case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A082550, A135342, A240851, A327475, A327482, A328966, A359906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Mean[#]!=Median[#]&]],{n,0,30}]

A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

Original entry on oeis.org

1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882

Offset: 1

Gus Wiseman, Feb 10 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).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    9: {2,2}
   54: {1,2,2,2}
  100: {1,1,3,3}
  120: {1,1,1,2,3}
  125: {3,3,3}
  135: {2,2,2,3}
  168: {1,1,1,2,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.

For mean instead of median we have A359903, counted by A360068.
For distinct indices instead of indices we have A360453, counted by A360455.
For distinct indices instead of multiplicities: A360249, counted by A360245.
These partitions are counted by A360456.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranked by A359889.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median.
A359894 counts partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
Cf. A000975, A109297, A109298, A114638, A316413, A324570, A360244, A360248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Median[prix[#]]==Median[Length/@Split[prix[#]]]&]

A360686 Number of integer partitions of n whose distinct parts have integer median.

Original entry on oeis.org

1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408

Offset: 1

Gus Wiseman, Feb 20 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) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (421)      (53)
                    (1111)           (51)      (511)      (62)
                                     (222)     (3211)     (71)
                                     (321)     (31111)    (422)
                                     (3111)    (1111111)  (431)
                                     (111111)             (521)
                                                          (2222)
                                                          (3221)
                                                          (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).

For all parts: A325347, strict A359907, ranks A359908, complement A307683.
For mean instead of median: A360241, ranks A326621.
These partitions have ranks A360550, complement A360551.
For multiplicities instead of distinct parts: A360687.
The complement is counted by A360689.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A027193 counts odd-length partitions, strict A067659, ranks A026424.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A240219, A359906, A360005, A360071, A360244, A360245, A360556, A360688.

Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Union[#]]]&]],{n,30}]

A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 50, 12, 14, 33, 54, 0, 115, 0, 92, 75, 31, 99, 323, 0, 45, 162, 443, 0, 507, 0, 467, 732, 88, 0, 1551, 274, 833, 627, 1228, 0, 2035, 1556, 2859, 1152, 221, 0, 9008, 0, 295, 4835, 5358

Offset: 1

Gus Wiseman, Jun 23 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 of {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 a(8) = 1 through a(18) = 12 partitions:
  3221  .  32221  .  4332    .  3222221  43332  5443      .  433332
                     5331       3322211  53331  6442         443331
                     322221     4222211  63321  7441         533322
                     422211                     32222221     533331
                                                33222211     543321
                                                42222211     633321
                                                52222111     733311
                                                             322222221
                                                             332222211
                                                             422222211
                                                             432222111
                                                             522222111

Non-constant partitions are counted by A144300, ranks A024619.
This is the non-constant case of A363719, ranks A363727.
These partitions have ranks A363729.
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, A363720, A363721.

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

A359891 Members of A026424 (numbers with an odd number of prime factors) whose prime indices have the same mean as median.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 30, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 110, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Gus Wiseman, Jan 22 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).

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   8: {1,1,1}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  23: {9}
  27: {2,2,2}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is not in the sequence.

A subset of A026424 = numbers with odd bigomega.
The LHS (mean of prime indices) is A326567/A326568.
This is the odd-length case of A359889, complement A359890.
The complement is A359892.
These partitions are counted by A359895, any-length A240219.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359893 and A359901 count partitions by median, odd-length A359902.
A359908 lists numbers whose prime indices have integer median.
Cf. A327473, A359894, A359899, A359910, A360007, A360009.

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

Intersection of A026424 and A359889.

A360674 Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 9, 16, 15, 21, 16, 34, 22, 33, 36, 47, 36, 62, 44, 75, 68, 78, 68, 120, 93, 113, 117, 151, 122, 195, 148, 209, 197, 220, 226, 315, 249, 304, 309, 402, 332, 463, 387, 496, 515, 539, 514, 712, 609, 738, 723, 845, 774, 983, 914, 1111

Offset: 0

Gus Wiseman, Mar 04 2023

nonn

Of course, only one of the two conditions is necessary.

			The a(1) = 1 through a(6) = 12 partitions:
  (11)  (22)    (33)      (44)        (55)          (66)
        (211)   (321)     (422)       (532)         (633)
        (1111)  (21111)   (431)       (541)         (642)
                (111111)  (2222)      (32221)       (651)
                          (22211)     (211111111)   (3333)
                          (2111111)   (1111111111)  (33222)
                          (11111111)                (33321)
                                                    (42222)
                                                    (222222)
                                                    (2222211)
                                                    (21111111111)
                                                    (111111111111)
For example, the partition y = (3,2,2,2,1) has halves (3,2) and (2,2,1), both with sum 5, so y is counted under a(5).

The even-length case is A000005.
Central diagonal of A360672.
These partitions have ranks A360953.
A008284 counts partitions by length, row sums A000041.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A237363, A360254, A360671, A360673, A360682.

Table[Length[Select[IntegerPartitions[2n], Total[Take[#,Floor[Length[#]/2]]]==n&]],{n,0,15}]

def accel_asc(n):
    a = [0 for i in range(n + 1)]
    k = 1
    y = n - 1
    while k != 0:
        x = a[k - 1] + 1
        k -= 1
        while 2 * x <= y:
            a[k] = x
            y -= x
            k += 1
        l = k + 1
        while x <= y:
            a[k] = x
            a[l] = y
            yield a[:k + 2]
            x += 1
            y -= 1
        a[k] = x + y
        y = x + y - 1
        yield a[:k + 1]
for y in range(1000):
    num = 0
    for x in accel_asc(2*y):
        stop = len(x)//2+1
        if len(x) % 2 == 0:
            stop -= 1
        right = x[0:stop]
        left = x[stop:]
        if sum(right) == sum(left):
            num += 1
    print(y,num)
# David Consiglio, Jr., Mar 09 2023

a(n) = A360672(2n,n).

More terms from David Consiglio, Jr., Mar 09 2023

cp's OEIS Frontend

A359909 Number of integer factorizations of n into factors > 1 with the same mean as median.

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A360249 Numbers for which the prime indices have the same median as the distinct prime indices.

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A360552 Numbers > 1 whose distinct prime factors have integer median.

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A359896 Number of odd-length integer partitions of n whose parts do not have the same mean as median.

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A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

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A360454 Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.

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A360686 Number of integer partitions of n whose distinct parts have integer median.

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A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A359891 Members of A026424 (numbers with an odd number of prime factors) whose prime indices have the same mean as median.

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