A326621 -id:A326621 - cp's OEIS frontend

A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

0, 0, 0, 0, 1, 3, 3, 9, 11, 19, 25, 43, 49, 82, 103, 136, 183, 258, 314, 435, 524, 687, 892, 1150, 1378, 1788, 2241, 2773, 3399, 4308, 5142, 6501, 7834, 9600, 11726, 14099, 16949, 20876, 25042, 30032, 35732, 43322, 51037, 61650, 72807, 86319, 102983, 122163

Offset: 0

Gus Wiseman, Feb 04 2023

nonn

			The a(1) = 0 through a(9) = 19 partitions:
  .  .  .  (211)  (221)   (411)    (322)     (332)      (441)
                  (311)   (3111)   (331)     (422)      (522)
                  (2111)  (21111)  (511)     (611)      (711)
                                   (2221)    (4211)     (3222)
                                   (3211)    (5111)     (3321)
                                   (4111)    (22211)    (4221)
                                   (22111)   (32111)    (4311)
                                   (31111)   (41111)    (5211)
                                   (211111)  (221111)   (6111)
                                             (311111)   (22221)
                                             (2111111)  (32211)
                                                        (33111)
                                                        (42111)
                                                        (51111)
                                                        (321111)
                                                        (411111)
                                                        (2211111)
                                                        (3111111)
                                                        (21111111)
For example, the partition y = (32211) has mean 9/5 and distinct parts {1,2,3} with mean 2, so y is counted under a(9).

The complement for multiplicities instead of distinct parts is A360068.
The complement is counted by A360243, ranks A360247.
For median instead of mean we have A360244, complement A360245.
These partitions have ranks A360246.
Sum of A360250 and A360251, ranks A360252 and 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.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156.

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

A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 11, 17, 13, 28, 19, 32, 40, 48, 39, 71, 55, 103, 105, 110, 105, 197, 170, 195, 237, 319, 257, 462, 341, 515, 543, 584, 784, 1028, 761, 973, 1153, 1606, 1261, 2137, 1611, 2368, 2815, 2575, 2591, 4393, 3798, 4602, 4663, 5777, 5121

Offset: 0

Gus Wiseman, Feb 04 2023

nonn

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (321)     (1111111)  (431)
                                     (2211)               (521)
                                     (111111)             (2222)
                                                          (3221)
                                                          (3311)
                                                          (11111111)

For multiplicities instead of distinct parts we have A360068.
The complement is counted by A360242, ranks A360246.
For median instead of mean we have A360245, complement A360244.
These partitions have ranks A360247.
Cf. A360250 and A360251, ranks A360252 and 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.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156, A360069.

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

A360551 Numbers > 1 whose distinct prime indices have non-integer median.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A325700 in having 330 and lacking 462.
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. Distinct prime indices are listed by A304038.
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 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.

For mean instead of median we have the complement of A326621.
Positions of odd terms in A360457.
The complement (without 1) is A360550, counted by A360686.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A348551, A360006, A360009, A360248, A360453.

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

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A187039 in having 2520 and lacking 1 and 12600.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
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 unordered prime signature of 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.

A subset of A030231.
For mean instead of median we have A070011.
Positions of odd terms in A360460.
The complement is A360553 (without 1), counted by A360687.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A304038, A316413, A326621, A348551, A360006, A360009, A360248, A360453.

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

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

Original entry on oeis.org

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[#]]]&]

A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 100, 112, 125, 180, 250, 252, 300, 352, 360, 392, 396, 405, 450, 468, 504, 540, 588, 600, 612, 675, 684, 720, 756, 792, 828, 832, 882, 900, 936, 1008, 1044, 1116, 1125, 1176, 1188, 1200, 1224, 1332, 1350, 1368, 1372, 1404, 1440, 1452, 1476

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}
   12: {1,1,2}
   18: {1,2,2}
   40: {1,1,1,3}
  100: {1,1,3,3}
  112: {1,1,1,1,4}
  125: {3,3,3}
  180: {1,1,2,2,3}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  300: {1,1,2,3,3}
  352: {1,1,1,1,1,5}
  360: {1,1,1,2,2,3}
For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.

Without taking median we have A109298, unordered A109297.
For mean instead of median we have A324570, counted by A114638.
For indices instead of multiplicities we have A360249, counted by A360245.
For indices instead of distinct indices we have A360454, counted by A360456.
These partitions are counted by A360455.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranks A359889.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A325347 = partitions with 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.
A360005 gives median of prime indices (times two).
Cf. A000975, A359890, A359903, A360068, A360244, A360247, A360248.

Select[Range[100],#==1||Median[Last/@FactorInteger[#]]== Median[PrimePi/@First/@FactorInteger[#]]&]

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A067340 in having 60.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
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 unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- 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.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

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

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).

cp's OEIS Frontend

A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

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A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

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A360551 Numbers > 1 whose distinct prime indices have non-integer median.

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A360554 Numbers > 1 whose unordered prime signature has non-integer median.

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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

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A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

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Mathematica

A360553 Numbers > 1 whose unordered prime signature has integer median.

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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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Mathematica