A327482 -id:A327482 - cp's OEIS frontend

A360246 Numbers for which the prime indices do not have the same mean as the distinct prime indices.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

First differs from A242416 in having 126.
Contains no squarefree numbers or perfect powers.
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:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with mean 9/4 and distinct prime indices {1,2,4} with mean 7/3, so 126 is in the sequence.

Signature instead of parts: complement A324570, counted by A114638.
Signature instead of distinct parts: complement A359903, counted by A360068.
These partitions are counted by A360242.
The complement is A360247, counted by A360243.
For median we have A360248, counted by A360244 (complement A360245).
Union of A360252 and A360253, counted by A360250 and A360251.
A058398 counts partitions by mean, also A327482.
A088529/A088530 gives mean of prime signature (A124010).
A112798 lists prime indices, length A001222, sum A056239.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
Cf. A000975, A051293, A067340, A067538, A360005, A360241.

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

A360247 Numbers for which the prime indices have the same mean 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, 127, 128, 129, 130

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

First differs from A072774 in having 90.
First differs from A242414 in lacking 126.
Includes all squarefree numbers and perfect powers.
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 prime indices of 900 are {3,3,2,2,1,1} with mean 2, and the distinct prime indices are {1,2,3} also with mean 2, so 900 is in the sequence.

Signature instead of parts: A324570, counted by A114638.
Signature instead of distinct parts: A359903, counted by A360068.
These partitions are counted by A360243.
The complement is A360246, counted by A360242.
For median instead of mean the complement is A360248, counted by A360244.
For median instead of mean we have A360249, counted by A360245.
For greater instead of equal mean we have A360252, counted by A360250.
For lesser instead of equal mean we have A360253, counted by A360251.
A008284 counts partitions by number of parts, distinct A116608.
A058398 counts partitions by mean, also A327482.
A088529/A088530 gives mean of prime signature (A124010).
A112798 lists prime indices, length A001222, sum A056239.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
Cf. A000975, A051293, A067340, A067538, A360005, A360241.

isA360247 := proc(n)
    local ifs,pidx,pe,meanAll,meanDist ;
    if n = 1 then
        return true ;
    end if ;
    ifs := ifactors(n)[2] ;
    # list of prime indices with multiplicity
    pidx := [] ;
    for pe in ifs do
        [numtheory[pi](op(1,pe)),op(2,pe)] ;
        pidx := [op(pidx),%] ;
    end do:
    meanAll := add(op(1,pe)*op(2,pe),pe=pidx) / add(op(2,pe),pe=pidx) ;
    meanDist := add(op(1,pe),pe=pidx) / nops(pidx) ;
    if meanAll = meanDist then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 130 do
    if isA360247(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],Mean[prix[#]]==Mean[Union[prix[#]]]&]

A360687 Number of integer partitions of n whose multiplicities have integer median.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676

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)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A362559 Number of integer partitions of n whose weighted sum is divisible by n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, 14, 18, 25, 28, 26, 42, 47, 52, 73, 77, 100, 118, 122, 158, 188, 219, 266, 313, 367, 412, 489, 578, 698, 809, 914, 1094, 1268, 1472, 1677, 1948, 2305, 2656, 3072, 3527, 4081, 4665, 5342, 6225, 7119, 8150, 9408

Offset: 1

Gus Wiseman, Apr 24 2023

nonn

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

			The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)     (7)        (8)       (9)
            (111)       (11111)  (222)   (3211)     (3311)    (333)
                                 (3111)  (1111111)  (221111)  (4221)
                                                              (222111)
                                                              (111111111)

For median instead of mean we have A362558.
The complement is counted by A362560.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.

Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A363948 Numbers whose prime indices have mean < 3/2.

Original entry on oeis.org

2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 432, 448, 480, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3456, 3584, 3840, 4096, 4608

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 initial terms, prime indices, and means:
    2: {1} -> 1
    4: {1,1} -> 1
    8: {1,1,1} -> 1
   12: {1,1,2} -> 4/3
   16: {1,1,1,1} -> 1
   24: {1,1,1,2} -> 5/4
   32: {1,1,1,1,1} -> 1
   48: {1,1,1,1,2} -> 6/5
   64: {1,1,1,1,1,1} -> 1
   72: {1,1,1,2,2} -> 7/5
   80: {1,1,1,1,3} -> 7/5
   96: {1,1,1,1,1,2} -> 7/6

These partitions are counted by A363947.
Prime indices have mean A326567/A326568.
For low mode we have A360015, high A360013.
Positions of 1's in A363489.
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.
A363949 ranks partitions with low mean 1, counted by A025065.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124944, A327473, A327476, A327482, A359889, A363727, A363942, A363943, A363946, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]<3/2&]

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

Original entry on oeis.org

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

A360008 Positions of first appearances in the sequence giving the mean of prime indices (A326567/A326568).

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 12, 13, 14, 17, 18, 19, 23, 24, 26, 29, 31, 37, 38, 41, 42, 43, 47, 48, 52, 53, 54, 58, 59, 61, 67, 71, 72, 73, 74, 76, 79, 83, 86, 89, 92, 96, 97, 101, 103, 104, 106, 107, 108, 109, 113, 122, 124, 127, 131, 137, 139, 142, 148, 149, 151, 152

Offset: 1

Gus Wiseman, Jan 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.

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   24: {1,1,1,2}

Positions of first appearances in A326567/A326568.
The version for median instead of mean is A360007, unsorted A360006.
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.
A326567/A326568 gives mean of prime indices.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A051293, A327473, A348551, A359889, A360005.

nn=1000;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,Mean[prix[n]]],{n,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

A360250 Number of integer partitions of n where the parts have greater mean than the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 2, 3, 3, 9, 5, 13, 15, 18, 20, 37, 34, 59, 51, 68, 92, 134, 121, 167, 203, 251, 282, 387, 375, 537, 561, 714, 888, 958, 1042, 1408, 1618, 1939, 2076, 2650, 2764, 3479, 3863, 4431, 5387, 6520, 6688, 8098, 9041, 10614, 12084, 14773, 15469

Offset: 0

Gus Wiseman, Feb 06 2023

nonn

			The a(5) = 1 through a(12) = 5 partitions:
  (221)  .  (331)   (332)    (441)    (442)     (443)      (552)
            (2221)  (22211)  (3321)   (3331)    (551)      (4431)
                             (22221)  (222211)  (3332)     (33321)
                                                (4331)     (44211)
                                                (4421)     (2222211)
                                                (33221)
                                                (33311)
                                                (222221)
                                                (2222111)
For example, the partition y = (4,3,3,1) has mean 11/4 and distinct parts {1,3,4} with mean 8/5, so y is counted under a(11).

For unequal instead of greater we have A360242, ranks A360246.
For equal instead of greater we have A360243, ranks A360247.
For less instead of greater we have A360251, ranks A360253.
These partitions have ranks A360252.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A240219 counts partitions with mean equal to median, ranks A359889.
A359894 counts partitions with mean different from median, ranks A359890.
A360071 counts partitions by number of parts and number of distinct parts.
Cf. A000975, A316313, A326567/A326568, A326619/A326620, A326621, A360068, A360241, A360244, A360245.

Table[Length[Select[IntegerPartitions[n],Mean[#]>Mean[Union[#]]&]],{n,0,30}]

a(n) + A360251(n) = A360242(n).

a(n) + A360251(n) + A360243(n) = A000041(n).

A360251 Number of integer partitions of n where the parts have lesser mean than the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 7, 9, 16, 22, 34, 44, 69, 88, 118, 163, 221, 280, 376, 473, 619, 800, 1016, 1257, 1621, 2038, 2522, 3117, 3921, 4767, 5964, 7273, 8886, 10838, 13141, 15907, 19468, 23424, 28093, 33656, 40672, 48273, 58171, 68944, 81888, 97596, 115643

Offset: 0

Gus Wiseman, Feb 06 2023

nonn

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (3111)   (511)     (611)      (711)
                 (21111)  (3211)    (4211)     (3222)
                          (4111)    (5111)     (4221)
                          (22111)   (32111)    (4311)
                          (31111)   (41111)    (5211)
                          (211111)  (221111)   (6111)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,2,1) has mean 9/4 and distinct parts {1,2,4} with mean 7/3, so y is counted under a(9).

For unequal instead of less we have A360242, ranks A360246.
For equal instead of less we have A360243, ranks A360247.
For greater instead of less we have A360250, ranks A360252.
These partitions have ranks A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A240219 counts partitions with mean equal to median, ranks A359889.
A359894 counts partitions with mean different from median, ranks A359890.
A360071 counts partitions by number of parts and number of distinct parts.
Cf. A000975, A316313, A326567/A326568, A326619/A326620, A326621, A360068, A360241, A360244, A360245.

Table[Length[Select[IntegerPartitions[n],Mean[#]

a(n) + A360250(n) = A360242(n).

a(n) + A360250(n) + A360243(n) = A000041(n).

A361906 Number of integer partitions of n such that (length) * (maximum) >= 2*n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 5, 9, 15, 19, 36, 43, 68, 96, 125, 171, 232, 297, 418, 529, 676, 853, 1156, 1393, 1786, 2316, 2827, 3477, 4484, 5423, 6677, 8156, 10065, 12538, 15121, 17978, 22091, 26666, 32363, 38176, 46640, 55137, 66895, 79589, 92621, 111485, 133485

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also partitions such that (maximum) >= 2*(mean).

These are partitions whose complement (see example) has size >= n.

			The a(6) = 2 through a(10) = 15 partitions:
  (411)   (511)    (611)     (621)      (721)
  (3111)  (4111)   (4211)    (711)      (811)
          (31111)  (5111)    (5211)     (5221)
                   (41111)   (6111)     (5311)
                   (311111)  (42111)    (6211)
                             (51111)    (7111)
                             (321111)   (42211)
                             (411111)   (43111)
                             (3111111)  (52111)
                                        (61111)
                                        (421111)
                                        (511111)
                                        (3211111)
                                        (4111111)
                                        (31111111)
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 >= 2*8, so y is counted under a(8).
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not >= 2*7, so y is not counted under a(7).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement (shown in dots) of size 5, and 5 is not >= 7, so y is not counted under a(7).

For length instead of mean we have A237752, reverse A237755.
For minimum instead of mean we have A237821, reverse A237824.
For median instead of mean we have A361859, reverse A361848.
The unequal case is A361907.
The complement is counted by A361852.
The equal case is A361853, ranks A361855.
Reversing the inequality gives A361851.
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, strict A102627, ranks A316413.
A268192 counts partitions by complement size, ranks A326844.
Cf. A027193, A111907, A116608, A237984, A324521, A327482, A349156, A360068.

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

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A360246 Numbers for which the prime indices do not have the same mean as the distinct prime indices.

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A360247 Numbers for which the prime indices have the same mean as the distinct prime indices.

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Maple

Mathematica

A360687 Number of integer partitions of n whose multiplicities have integer median.

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Mathematica

A362559 Number of integer partitions of n whose weighted sum is divisible by n.

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Mathematica

A363948 Numbers whose prime indices have mean < 3/2.

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Mathematica

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

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PARI

Extensions

A360008 Positions of first appearances in the sequence giving the mean of prime indices (A326567/A326568).

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Mathematica

A360250 Number of integer partitions of n where the parts have greater mean than the distinct parts.

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Formula

A360251 Number of integer partitions of n where the parts have lesser mean than the distinct parts.

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