A360250 -id:A360250

A360244

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

+10
18

0, 0, 0, 0, 1, 3, 3, 9, 11, 17, 23, 37, 42, 68, 87, 110, 153, 209, 261, 352, 444, 573, 750, 949, 1187, 1508, 1909, 2367, 2938, 3662, 4507, 5576, 6826, 8359, 10203, 12372, 15011, 18230, 21996, 26518, 31779, 38219, 45682, 54660, 65112, 77500, 92089, 109285

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

LINKS

Table of n, a(n) for n=0..47.

EXAMPLE

The a(4) = 1 through a(9) = 17 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) (4311)

(22111) (32111) (5211)

(31111) (41111) (6111)

(211111) (221111) (22221)

(311111) (33111)

(2111111) (42111)

(51111)

(321111)

(411111)

(2211111)

(3111111)

(21111111)

For example, the partition y = (33111) has median 1, and the distinct parts {1,3} have median 2, so y is counted under a(9).

MATHEMATICA

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

CROSSREFS

For mean instead of median: A360242, ranks A360246, complement A360243.

These partitions are ranked by A360248.

The complement is A360245, ranked by A360249.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A116608 counts partitions by number of distinct parts.

A240219 counts partitions with mean equal to median, ranks A359889.

A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

A359893 and A359901 count partitions by median.

A359894 counts partitions with mean different from median, ranks A359890.

A360071 counts partitions by number of parts and number of distinct parts.

Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360068, A360241, A360250, A360251.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 05 2023

STATUS

approved

A360245

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

+10
17

1, 1, 2, 3, 4, 4, 8, 6, 11, 13, 19, 19, 35, 33, 48, 66, 78, 88, 124, 138, 183, 219, 252, 306, 388, 450, 527, 643, 780, 903, 1097, 1266, 1523, 1784, 2107, 2511, 2966, 3407, 4019, 4667, 5559, 6364, 7492, 8601, 10063, 11634, 13469, 15469, 17985, 20558, 23812

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

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 example, the partition y = (6,4,4,4,1,1) has median 4, and the distinct parts {1,4,6} also have median 4, so y is counted under a(20).

MATHEMATICA

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

CROSSREFS

For mean instead of median: A360242, ranks A360247, complement A360243.

These partitions have ranks A360249.

The complement is A360244, ranks A360248.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A116608 counts partitions by number of distinct parts.

A240219 counts partitions with mean equal to median, ranks A359889.

A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

A359893 and A359901 count partitions by median.

A359894 counts partitions with mean different from median, ranks A359890.

A360071 counts partitions by number of parts and number of distinct parts.

Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360241, A360246, A360250, A360251.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 05 2023

STATUS

approved

A360241

Number of integer partitions of n whose distinct parts have integer mean.

+10
16

0, 1, 2, 2, 4, 3, 8, 6, 13, 13, 22, 19, 43, 34, 56, 66, 97, 92, 156, 143, 233, 256, 322, 341, 555, 542, 710, 831, 1098, 1131, 1644, 1660, 2275, 2484, 3035, 3492, 4731, 4848, 6063, 6893, 8943, 9378, 12222, 13025, 16520, 18748, 22048, 24405, 31446, 33698, 41558

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

The a(1) = 1 through a(8) = 13 partitions:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (111) (22) (311) (33) (331) (44)

(31) (11111) (42) (511) (53)

(1111) (51) (3211) (62)

(222) (31111) (71)

(321) (1111111) (422)

(3111) (2222)

(111111) (3221)

(3311)

(5111)

(32111)

(311111)

(11111111)

For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).

MATHEMATICA

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

CROSSREFS

For parts instead of distinct parts we have A067538, ranked by A316413.

The strict case is A102627.

These partitions are ranked by A326621.

For multiplicities instead of distinct parts: A360069, ranked by A067340.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A051293 counts subsets with integer mean, median A000975.

A058398 counts partitions by mean, also A327482.

A116608 counts partitions by number of distinct parts.

A326619/A326620 gives mean of distinct prime indices.

A326622 counts factorizations with integer mean, strict A328966.

A360071 counts partitions by number of parts and number of distinct parts.

The following count partitions:

- A360242 mean(parts) != mean(distinct parts), ranked by A360246.

- A360243 mean(parts) = mean(distinct parts), ranked by A360247.

- A360250 mean(parts) > mean(distinct parts), ranked by A360252.

- A360251 mean(parts) < mean(distinct parts), ranked by A360253.

Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 02 2023

STATUS

approved

A360242

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

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..47.

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 04 2023

STATUS

approved

A360243

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

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..53.

EXAMPLE

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)

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 04 2023

STATUS

approved

A360246

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

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.

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], Mean[prix[#]]!=Mean[Union[prix[#]]]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 07 2023

STATUS

approved

A360247

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..96.

EXAMPLE

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.

MAPLE

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

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], Mean[prix[#]]==Mean[Union[prix[#]]]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 07 2023

STATUS

approved

A360251

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..47.

FORMULA

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

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

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 06 2023

STATUS

approved

A360252

Numbers for which the prime indices have greater mean than the distinct prime indices.

+10
9

18, 50, 54, 75, 98, 108, 147, 150, 162, 242, 245, 250, 294, 324, 338, 350, 363, 375, 450, 486, 490, 500, 507, 578, 588, 605, 648, 686, 722, 726, 735, 750, 845, 847, 867, 882, 972, 1014, 1029, 1050, 1058, 1078, 1083, 1125, 1183, 1210, 1250, 1274, 1350, 1372

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..50.

EXAMPLE

The terms together with their prime indices begin:

18: {1,2,2}

50: {1,3,3}

54: {1,2,2,2}

75: {2,3,3}

98: {1,4,4}

108: {1,1,2,2,2}

147: {2,4,4}

150: {1,2,3,3}

162: {1,2,2,2,2}

242: {1,5,5}

245: {3,4,4}

250: {1,3,3,3}

294: {1,2,4,4}

324: {1,1,2,2,2,2}

For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is in the sequence.

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], Mean[prix[#]]>Mean[Union[prix[#]]]&]

CROSSREFS

For unequal instead of greater we have A360246, counted by A360242.

For equal instead of greater we have A360247, counted by A360243.

These partitions are counted by A360250.

For less instead of greater we have A360253, counted by A360251.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose indices have integer mean, distinct A326621.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

Cf. A000975, A051293, A058398, A067340, A067538, A324570, A327482, A359903, A360005, A360241, A360248.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 09 2023

STATUS

approved

A360253

Numbers for which the prime indices have lesser mean than the distinct prime indices.

+10
9

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212, 220

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

The terms together with their prime indices begin:

12: {1,1,2}

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}

52: {1,1,6}

56: {1,1,1,4}

60: {1,1,2,3}

63: {2,2,4}

68: {1,1,7}

72: {1,1,1,2,2}

For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is not in the sequence.

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], Mean[prix[#]]<Mean[Union[prix[#]]]&]

CROSSREFS

These partitions are counted by A360251.

For unequal instead of less we have A360246, counted by A360242.

For equal instead of less we have A360247, counted by A360243.

For greater instead of less we have A360252, counted by A360250.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose indices have integer mean, distinct A326621.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

Cf. A000975, A051293, A058398, A067340, A067538, A324570, A327482, A359903, A360005, A360241, A360248.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 09 2023

STATUS

approved