A326619 -id:A326619 - cp's OEIS frontend

A360457 Two times the median of the set of distinct prime indices of n; a(1) = 1.

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

Offset: 1

Gus Wiseman, Feb 14 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). 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.

			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.

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.

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

A328966 Number of strict factorizations of n with integer average.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Nov 16 2019

nonn

			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)

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.

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}]

A326620 Denominator of the average of the set of distinct prime indices of n.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Jul 14 2019

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 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.

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences related to prime indices in the factorization of n.

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.

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

A326620(n) = if(1==n,0,denominator(vecsum(apply(primepi,factor(n)[,1]))/omega(n))); \\ Antti Karttunen, Jan 28 2025

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 14 2019

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 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.

			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}

Positions of 1's in A326620.

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

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

A360248 Numbers for which the prime indices do not have the same median 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, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 200

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

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).

			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.

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.

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

A360244 Number of integer partitions of n where the parts do not have the same median as the distinct parts.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 05 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(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).

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.

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

A360550 Numbers > 1 whose distinct prime indices have integer median.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 14 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. 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 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.

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.

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

A360245 Number of integer partitions of n where the parts have the same median as the distinct parts.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 05 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) = 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).

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.

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

A123529 Denominator of average of prime factors of n.

Original entry on oeis.org

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

Offset: 2

Franklin T. Adams-Watters, Oct 02 2006

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

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

See A123528 for more formulas and references.

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

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

A360241 Number of integer partitions of n whose distinct parts have integer mean.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 02 2023

nonn

			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).

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.

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

cp's OEIS Frontend

A360457 Two times the median of the set of distinct prime indices of n; a(1) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A328966 Number of strict factorizations of n with integer average.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A326620 Denominator of the average of the set of distinct prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360248 Numbers for which the prime indices do not have the same median as the distinct prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360244 Number of integer partitions of n where the parts do not have the same median as the distinct parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360550 Numbers > 1 whose distinct prime indices have integer median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360245 Number of integer partitions of n where the parts have the same median as the distinct parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A123529 Denominator of average of prime factors of n.

Views

Author

Keywords

Comments