A326619 -id:A326619

A360457

Two times the median of the set of distinct prime indices of n; a(1) = 1.

+10
40

1, 2, 4, 2, 6, 3, 8, 2, 4, 4, 10, 3, 12, 5, 5, 2, 14, 3, 16, 4, 6, 6, 18, 3, 6, 7, 4, 5, 20, 4, 22, 2, 7, 8, 7, 3, 24, 9, 8, 4, 26, 4, 28, 6, 5, 10, 30, 3, 8, 4, 9, 7, 32, 3, 8, 5, 10, 11, 34, 4, 36, 12, 6, 2, 9, 4, 38, 8, 11, 6, 40, 3, 42, 13, 5, 9, 9, 4, 44, 4

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

OFFSET

1,2

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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

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.

LINKS

Table of n, a(n) for n=1..80.

EXAMPLE

The prime indices of 65 are {3,6}, with distinct parts {3,6}, with median 9/2, so a(65) = 9.

The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so a(900) = 4.

MATHEMATICA

Table[If[n==1, 1, 2*Median[PrimePi/@First/@FactorInteger[n]]], {n, 100}]

CROSSREFS

The version for divisors is A063655.

For mean instead of two times median we have A326619/A326620.

The version for all prime indices is A360005.

Positions of first appearances are A360006, sorted A360007.

The version for distinct prime factors is A360458.

The version for all prime factors is A360459.

The version for prime multiplicities is A360460.

Positions of even terms are A360550.

Positions of odd terms are A360551.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

A304038 lists distinct prime indices.

A325347 counts partitions with integer median, complement A307683.

A326567/A326568 gives mean of prime indices.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A000975, A026424, A316413, A359907, A359908, A359912, A360248, A360453.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 14 2023

STATUS

approved

A326620

Denominator of the average of the set of distinct prime indices of n.

+10
27

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1

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

OFFSET

2,5

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=2..88.

EXAMPLE

The distinct prime indices of 12 are {1,2}, with average 3/2, so a(12) = 2.

The sequence of fractions begins: 1, 2, 1, 3, 3/2, 4, 1, 2, 2, 5, 3/2, 6, 5/2, 5/2, 1, 7, 3/2, 8, 2, 3, 3, 9, 3/2, 3, 7/2, 2, 5/2, 10, 2.

MATHEMATICA

Table[Denominator[Mean[PrimePi/@First/@FactorInteger[n]]], {n, 2, 100}]

CROSSREFS

Positions of 1's are A326621.

The average of the multiset of prime indices is A326567/A326568.

The average of the multiset of prime factors is A123528/A123529.

The average of the set of distinct prime indices is A326619/A326620.

The average of the set of distinct prime factors is A323171/A323172.

Cf. A001221, A067629, A078174, A078175, A112798, A316413.

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jul 14 2019

STATUS

approved

A326621

Numbers n such that the average of the set of distinct prime indices of n is an integer.

+10
25

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 67, 68, 71, 73, 78, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100, 101, 103, 105

(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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts have an integer average.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

The sequence of terms together with their prime indices begins:

2: {1}

3: {2}

4: {1,1}

5: {3}

7: {4}

8: {1,1,1}

9: {2,2}

10: {1,3}

11: {5}

13: {6}

16: {1,1,1,1}

17: {7}

19: {8}

20: {1,1,3}

21: {2,4}

22: {1,5}

23: {9}

25: {3,3}

27: {2,2,2}

29: {10}

MATHEMATICA

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

CROSSREFS

Positions of 1's in A326620.

Cf. A051293, A056239, A067538, A078174, A078175, A102627, A112798, A326567/A326568, A326619/A326620, A326622.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 14 2019

STATUS

approved

A360248

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

+10
24

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, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 200

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

OFFSET

1,1

COMMENTS

First differs from A242416 in lacking 180, with prime indices {1,1,2,2,3}.

First differs from A360246 in lacking 126 and having 1950.

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).

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 median 2 and distinct prime indices {1,2,4} with median 2, so 126 is not in the sequence.

The prime indices of 1950 are {1,2,3,3,6} with median 3 and distinct prime indices {1,2,3,6} with median 5/2, so 1950 is in the sequence.

MATHEMATICA

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

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

CROSSREFS

These partitions are counted by A360244.

The complement is A360249, counted by A360245.

For multiplicities instead of parts: complement of A360453.

For multiplicities instead of distinct parts: complement of A360454.

For mean instead of median we have A360246, counted by A360242.

The complement for mean instead of median is A360247, counted by A360243.

A112798 lists prime indices, length A001222, sum A056239.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

A325347 = partitions with integer median, strict A359907, ranked by A359908.

A359893 and A359901 count partitions by median.

A360005 gives median of prime indices (times two).

Cf. A000975, A078174, A316413, A324570, A359890, A360455, A360456.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 07 2023

STATUS

approved

A328966

Number of strict factorizations of n with integer average.

+10
23

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 1, 5, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 5, 1, 1, 3, 3, 2, 1, 1, 2, 2, 1, 1, 5, 1, 1, 3, 2, 2, 2, 1, 5, 2, 1, 1, 4, 2, 1, 2, 3

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

OFFSET

2,7

LINKS

Table of n, a(n) for n=2..88.

EXAMPLE

The a(n) factorizations for n = 2, 8, 24, 48, 96:

(2) (8) (24) (32) (48) (96)

(2*4) (4*6) (4*8) (6*8) (2*48)

(2*12) (2*16) (2*24) (4*24)

(2*3*4) (4*12) (6*16)

(2*4*6) (8*12)

(3*4*8)

(2*3*16)

(2*4*12)

MATHEMATICA

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], UnsameQ@@#&&IntegerQ[Mean[#]]&]], {n, 2, 100}]

CROSSREFS

The non-strict version is A326622.

Partitions with integer average are A067538.

Strict partitions with integer average are A102627.

Heinz numbers of partitions with integer average are A316413.

Factorizations with integer geometric mean are A326028.

Cf. A001055, A051293, A078174, A078175, A326515, A326567/A326568, A326619/A326620, A326621, A326625.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 16 2019

STATUS

approved

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

A360550

Numbers > 1 whose distinct prime indices have integer median.

+10
18

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 73, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100

(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. 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).

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is in the sequence.

The prime indices of 330 are {1,2,3,5}, with distinct parts {1,2,3,5}, with median 5/2, so 330 is not in the sequence.

MATHEMATICA

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

CROSSREFS

For mean instead of median we have A326621.

Positions of even terms in A360457.

The complement (without 1) is A360551.

Partitions with these Heinz numbers are counted by A360686.

- 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.

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, A360006, A360009, A360248, A360453.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 14 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

A123529

Denominator of average of prime factors of n.

+10
16

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 3, 2, 1, 5, 1, 1, 1, 3, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 2, 3, 3, 1, 1, 1, 5, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 6, 1, 3, 3, 2, 1, 3, 1, 4, 1, 2

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

OFFSET

2,5

COMMENTS

Prime factors counted with multiplicity. - Harvey P. Dale, Jun 20 2013

Positions of 1's are A078175. a(n) is a divisor of Omega(n) = A001222(n). The average of prime indices (as opposed to prime factors) of n is A326567(n)/A326568(n). - Gus Wiseman, Jul 18 2019

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..5000

MATHEMATICA

Table[Denominator[Mean[Flatten[Table[#[[1]], {#[[2]]}]&/@ FactorInteger[ n]]]], {n, 110}] (* Harvey P. Dale, Jun 20 2013 *)

CROSSREFS

See A123528 for more formulas and references.

Cf. A078174, A078175, A112798, A316413, A326567/A326568, A326619/A326620, A326621.

KEYWORD

frac,nonn

AUTHOR

Franklin T. Adams-Watters, Oct 02 2006

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