A325347 -id:A325347 - cp's OEIS frontend

Previous Showing 41-50 of 107 results. Next

A363720 Number of integer partitions of n with different mean, median, and mode.

0, 0, 0, 0, 0, 0, 0, 2, 3, 5, 7, 16, 17, 34, 38, 50, 79, 115, 123, 198, 220, 291, 399, 536, 605, 815, 1036, 1241, 1520, 2059, 2315, 3132, 3708, 4491, 5668, 6587, 7788, 10259, 12299, 14515, 17153, 21558, 24623, 30876, 35540, 41476, 52023, 61931, 70811, 85545

Offset: 0

Gus Wiseman, Jun 21 2023

nonn

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(7) = 2 through a(11) = 16 partitions:
  (421)   (431)   (621)    (532)     (542)
  (3211)  (521)   (3321)   (541)     (632)
          (4211)  (4311)   (631)     (641)
                  (5211)   (721)     (731)
                  (32211)  (5311)    (821)
                           (6211)    (4322)
                           (322111)  (4421)
                                     (5321)
                                     (5411)
                                     (6311)
                                     (7211)
                                     (33221)
                                     (43211)
                                     (52211)
                                     (332111)
                                     (422111)

For equal instead of unequal: A363719, ranks A363727, odd-length A363721.
The case of a unique mode is A363725.
These partitions have ranks A363730.
For factorizations we have A363742, for equal A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363728.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,0,30}]

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

A000009(n) <= a(n) <= A000041(n).
Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - Clark Kimberling, Apr 04 2014
Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - Gus Wiseman, May 07 2023

			a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

For median instead of mode we have A053263.
The complement is counted by A240302.
The case where the maximum is the only mode is A362612.
A000041 counts integer partitions, strict A000009.
A362608 counts partitions with a unique mode, complement A362607.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes.
Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
(* Clark Kimberling, Apr 04 2014 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
     If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := PartitionsP[n] - b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz in A240302 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* Gus Wiseman, May 07 2023 *)

{ my(N=53, x='x+O('x^N));
my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
Vec(gf) } \\ John Tyler Rascoe, Mar 09 2024

a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j0 then 0 else 1).

a(n) + A240302(n) = A000041(n). - Clark Kimberling, Apr 04 2014.

G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - John Tyler Rascoe, Mar 09 2024

A231147 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) = (x^n - 1)/(x - 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 4, 9, 4, 5, 1, 1, 1, 1, 6, 5, 14, 9, 14, 5, 6, 1, 1, 1, 1, 7, 6, 20, 14, 29, 14, 20, 6, 7, 1, 1, 1, 1, 8, 7, 27, 20, 49, 29, 49, 20, 27, 7, 8, 1, 1, 1, 1, 9, 8, 35, 27, 76, 49, 99, 49, 76, 27, 35, 8, 9

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Clark Kimberling, Nov 05 2013

Keywords

nonn
tabf
easy

Comments

From Gus Wiseman, Mar 19 2023: (Start)

Also appears to be the number of nonempty subsets of {1,...,n} with median k, where k ranges from 1 to n in steps of 1/2, and the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:

{1} {1,2} {2} {1,4} {3} {2,5} {4} {4,5} {5}

{1,3} {2,3} {1,5} {3,4} {3,5}

{1,2,3} {1,2,3,4} {2,4} {1,3,4,5} {1,4,5}

{1,2,4} {1,2,3,5} {1,3,4} {2,3,4,5} {2,4,5}

{1,2,5} {1,3,5} {3,4,5}

{2,3,4}

{2,3,5}

{1,2,4,5}

{1,2,3,4,5}

Central diagonals T(n,(n+1)/2) appear to be A100066 (bisection A006134).

For mean instead of median we have A327481.

For partitions instead of subsets we have A359893, full steps A359901.

Central diagonals T(n,n/2) are A361801 (bisection A079309).

(End)

Examples

Triangle begins: 1 1 1 1 1 1 3 1 1 1 1 4 3 4 1 1 1 1 5 4 9 4 5 1 1 1 1 6 5 14 9 14 5 6 1 1 1 1 7 6 20 14 29 14 20 6 7 1 1 1 1 8 7 27 20 49 29 49 20 27 7 8 1 1 1 1 9 8 35 27 76 49 99 49 76 27 35 8 9 1 1 First 3 polynomials: 1, 1 + x + x^2, 1 + x + 3*x^2 + x^3 + x^4

Links

John Tyler Rascoe, Rows n = 1..100, flattened

Crossrefs

Cf. A231148.

Row sums are 2^n-1 = A000225(n).

Row lengths are 2n-1 = A005408(n-1).

Removing every other column appears to give A013580.

Cf. A006134, A007318, A008289, A079309, A325347, A327481, A359907, A360005.

Programs

Mathematica

z = 60; p[n_, x_] := p[x] = (x^n - 1)/(x - 1); Table[p[n, x], {n, 1, z/4}]; f1[n_, x_] := f1[n, x] = Numerator[Factor[p[n, x] /. x -> x + 1/x]]; Table[Expand[f1[n, x]], {n, 0, z/4}] Flatten[Table[CoefficientList[f1[n, x], x], {n, 1, z/4}]]

PARI

A231147_row(n) = {Vecrev(Vec(numerator((-1+(x+(1/x))^n)/(x+(1/x)-1))))} \\ John Tyler Rascoe, Sep 10 2024

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

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Feb 02 2023

Keywords

nonn

Examples

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).

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.

Programs

Mathematica

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

A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 6, 4, 2, 2, 3, 2, 2, 2, 8, 2, 3, 2, 3, 2, 2, 2, 4, 4, 2, 6, 3, 2, 2, 2, 10, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 4, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 3, 12, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 8, 2, 2, 2, 2, 2, 2

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Feb 14 2023

Keywords

nonn

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 number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

Examples

The unordered prime signature of 2520 is {1,1,2,3}, with median 3/2, so a(2520) = 3.

Crossrefs

The version for divisors is A063655.

For mean instead of two times median we have A088529/A088530.

Prime signature is A124010, unordered A118914.

The version for prime indices is A360005.

The version for distinct prime indices is A360457.

The version for distinct prime factors is A360458.

The version for prime factors is A360459.

Positions of even terms are A360553.

Positions of odd terms are A360554.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

A304038 lists distinct prime indices.

A325347 counts partitions w/ integer median, complement A307683.

A329976 counts partitions with median multiplicity 1.

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

Cf. A000975, A026424, A133464, A359907, A359908, A360454.

Programs

Mathematica

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

A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 3, 8, 8, 17, 19, 28, 39, 59, 68, 106, 123, 165, 220, 301, 361, 477, 605, 745, 929, 1245, 1456, 1932, 2328, 2846, 3590, 4292, 5111, 6665, 8040, 9607, 11532, 14410, 16699, 20894, 24287, 28706, 35745, 42845, 49548, 59963, 70985

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Jun 22 2023

Keywords

nonn

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).

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

The a(7) = 1 through a(13) = 17 partitions: (3211) (4211) (3321) (5311) (4322) (4431) (4432) (4311) (6211) (4421) (5322) (5422) (5211) (322111) (5411) (6411) (5521) (6311) (7311) (6322) (7211) (8211) (6511) (43211) (53211) (7411) (332111) (432111) (8311) (422111) (522111) (9211) (54211) (63211) (333211) (433111) (442111) (532111) (622111) (3322111) (32221111)

Crossrefs

The length-4 case appears to be A325695.

For equal instead of unequal we have A363719, ranks A363727.

Allowing multiple modes gives A363720, ranks A363730.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length (or decreasing mean), strict A008289.

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

A362608 counts partitions with a unique mode.

Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A359896, A359900, A363723, A363728.

Programs

Mathematica

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&]; Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]],{n,0,30}]

A359899 Number of strict odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 6, 1, 1, 6, 1, 5, 7, 1, 1, 8, 12, 1, 9, 2, 1, 33, 1, 1, 11, 1, 50, 12, 1, 1, 13, 70, 1, 46, 1, 1, 122, 1, 1, 16, 102, 155, 17, 1, 1, 30, 216, 258, 19, 1, 1, 310, 1, 1, 666, 1, 382, 23, 1, 1, 23, 1596, 1, 393, 1, 1

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Jan 20 2023

Keywords

nonn

Examples

The a(30) = 33 partitions: (30) (11,10,9) (8,7,6,5,4) (12,10,8) (9,7,6,5,3) (13,10,7) (9,8,6,4,3) (14,10,6) (9,8,6,5,2) (15,10,5) (10,7,6,4,3) (16,10,4) (10,7,6,5,2) (17,10,3) (10,8,6,4,2) (18,10,2) (10,8,6,5,1) (19,10,1) (10,9,6,3,2) (10,9,6,4,1) (11,7,6,4,2) (11,7,6,5,1) (11,8,6,3,2) (11,8,6,4,1) (11,9,6,3,1) (12,7,6,3,2) (12,7,6,4,1) (12,8,6,3,1) (12,9,6,2,1) (13,7,6,3,1) (13,8,6,2,1) (14,7,6,2,1) (11,10,6,2,1)

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Strict odd-length case of A240219, complement A359894, ranked by A359889.

Strict case of A359895, complement A359896, ranked by A359891.

Odd-length case of A359897, complement A359898.

The complement is counted by A359900.

A000041 counts partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.

A008289 counts strict partitions by mean.

A027193 counts odd-length partitions, strict A067659, ranked by A026424.

A067538 counts ptns with integer mean, strict A102627, ranked by A316413.

A237984 counts ptns containing their mean, strict A240850, ranked by A327473.

A325347 counts ptns with integer median, strict A359907, ranked by A359908.

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

Cf. A000016, A065795, A066571, A240851, A359903, A359906, A359910.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]

PARI

\\ Q(n,k,m) is g.f. for k strict parts of max size m. Q(n,k,m)={polcoef(prod(i=1, m, 1 + y*x^i + O(x*x^n)), k, y)} a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)); if(r>=h*(h+1), polcoef(Q(r, h, m-1)*Q(r, h, r), r)))))} \\ Andrew Howroyd, Jan 21 2023

Formula

a(p) = 1 for prime p. - Andrew Howroyd, Jan 21 2023

A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 4, 26, 9, 8, 4, 34, 6, 38, 4, 10, 13, 46, 4, 10, 15, 6, 4, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 4, 82, 6, 86, 4, 6, 25, 94, 4, 14, 10, 20, 4, 106, 6, 16, 4, 22, 31, 118, 5, 122, 33, 6, 4, 18, 6, 134, 4, 26, 10, 142, 4, 146

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Feb 14 2023

Keywords

nonn

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.

Examples

The prime factors of 60 are {2,2,3,5}, with median 5/2, so a(60) = 5.

Crossrefs

The union is 2 followed by A014091, complement of A014092.

The prime factors themselves are listed by A027746, distinct A027748.

The version for divisors is A063655.

Positions of odd terms are A072978 (except 1).

For mean instead of twice median: A123528/A123529, distinct A323171/A323172.

Positions of even terms are A359913 (and 1).

The version for prime indices is A360005.

The version for distinct prime indices is A360457.

The version for distinct prime factors is A360458.

The version for prime multiplicities is A360460.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

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, A027336, A078174, A316413, A359907, A359908, A360006, A360007, A360248, A360552.

Programs

Mathematica

Table[2*Median[Join@@ConstantArray@@@FactorInteger[n]],{n,100}]

A359906 Number of integer partitions of n with integer mean and integer median.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 10, 9, 14, 2, 39, 2, 24, 51, 49, 2, 109, 2, 170, 144, 69, 2, 455, 194, 116, 381, 668, 2, 1378, 2, 985, 956, 316, 2043, 4328, 2, 511, 2293, 6656, 2, 8634, 2, 8062, 14671, 1280, 2, 26228, 8035, 15991, 11614, 25055, 2, 47201, 39810, 65092

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Jan 21 2023

Keywords

nonn

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).

Examples

The a(1) = 1 through a(9) = 9 partitions: 1 2 3 4 5 6 7 8 9 11 111 22 11111 33 1111111 44 333 31 42 53 432 1111 51 62 441 222 71 522 321 2222 531 411 3221 621 111111 3311 711 5111 111111111 11111111

Crossrefs

For just integer mean we have A067538, strict A102627, ranked by A316413.

For just integer median we have A325347, strict A359907, ranked by A359908.

These partitions are ranked by A360009.

A000041 counts partitions, strict A000009.

A058398 counts partitions by mean, see also A008284, A327482.

A051293 counts subsets with integer mean, median A000975.

A326567/A326568 gives mean of prime indices.

A326622 counts factorizations with integer mean, strict A328966.

A359893/A359901/A359902 count partitions by median.

A360005(n)/2 gives median of prime indices.

Cf. A000016, A082550, A237984, A240219, A326669, A327475, A349156, A359894, A359897, A359905.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[#]]&&IntegerQ[Median[#]]&]],{n,1,30}]

A360458 Two times the median of the set of distinct prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 5, 26, 9, 8, 4, 34, 5, 38, 7, 10, 13, 46, 5, 10, 15, 6, 9, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 7, 82, 6, 86, 13, 8, 25, 94, 5, 14, 7, 20, 15, 106, 5, 16, 9, 22, 31, 118, 6, 122, 33, 10, 4, 18, 6, 134, 19, 26, 10, 142, 5

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Gus Wiseman, Feb 14 2023

Keywords

nonn

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.

Examples

The prime factors of 336 are {2,2,2,2,3,7}, with distinct parts {2,3,7}, with median 3, so a(336) = 6.

Crossrefs

The union is 2 followed by A014091, complement of A014092.

Distinct prime factors are listed by A027748.

The version for divisors is A063655.

Positions of odd terms are A100367.

For mean instead of two times median we have A323171/A323172.

The version for prime indices is A360005.

The version for distinct prime indices is A360457.

The version for prime factors is A360459.

The version for prime multiplicities is A360460.

Positions of even terms are A360552.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

A304038 lists distinct prime indices.

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

Cf. A000975, A026424, A078174, A316413, A325347, A359907, A360006, A360248, A360453, A360550, A360551.

Programs

Mathematica

Table[2*Median[First/@FactorInteger[n]],{n,100}]

Previous Showing 41-50 of 107 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
Content is available under The OEIS End-User License Agreement.

cp's OEIS Frontend

A363720 Number of integer partitions of n with different mean, median, and mode.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A231147 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) = (x^n - 1)/(x - 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359899 Number of strict odd-length integer partitions of n whose parts have the same mean as median.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359906 Number of integer partitions of n with integer mean and integer median.