cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A359893 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 3, 0, 1, 2, 0, 0, 0, 0, 1, 4, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 6, 1, 3, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 8, 1, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 11, 2, 7, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2023

Keywords

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

			Triangle begins:
  1
  1  0  1
  1  1  0  0  1
  2  0  2  0  0  0  1
  3  0  1  2  0  0  0  0  1
  4  1  2  0  3  0  0  0  0  0  1
  6  1  3  0  1  3  0  0  0  0  0  0  1
  8  1  6  0  2  0  4  0  0  0  0  0  0  0  1
 11  2  7  1  3  0  1  4  0  0  0  0  0  0  0  0  1
 15  2 10  3  4  0  2  0  5  0  0  0  0  0  0  0  0  0  1
 20  3 13  3  7  0  3  0  1  5  0  0  0  0  0  0  0  0  0  0  1
 26  4 19  3 11  1  4  0  2  0  6  0  0  0  0  0  0  0  0  0  0  0  1
For example, row n = 8 counts the following partitions:
  611       4211  422    .  332  .  44  .  .  .  .  .  .  .  8
  5111            521       431     53
  32111           2222              62
  41111           3221              71
  221111          3311
  311111          22211
  2111111
  11111111

Crossrefs

Row sums are A000041.
Row lengths are 2n-1 = A005408(n-1).
Column k=1 is A027336(n+1).
For mean instead of median we have A058398, see also A008284, A327482.
The mean statistic is ranked by A326567/A326568.
Omitting half-steps gives A359901.
The odd-length case is A359902.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
Cf. A325347, A349156, A359889, A359895, A359906, A359907, A360008.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Median[#]==k&]],{n,1,10},{k,1,n,1/2}]

A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 4, 3, 2, 0, 0, 0, 0, 1, 7, 4, 3, 1, 0, 0, 0, 0, 1, 8, 6, 3, 2, 0, 0, 0, 0, 0, 1, 12, 8, 4, 3, 1, 0, 0, 0, 0, 0, 1, 14, 11, 5, 4, 2, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2023

Keywords

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

			Triangle begins:
  1
  0  1
  1  0  1
  1  0  0  1
  2  1  0  0  1
  2  2  0  0  0  1
  4  2  1  0  0  0  1
  4  3  2  0  0  0  0  1
  7  4  3  1  0  0  0  0  1
  8  6  3  2  0  0  0  0  0  1
 12  8  4  3  1  0  0  0  0  0  1
 14 11  5  4  2  0  0  0  0  0  0  1
 21 14  8  4  3  1  0  0  0  0  0  0  1
 24 20 10  5  4  2  0  0  0  0  0  0  0  1
 34 25 15  6  5  3  1  0  0  0  0  0  0  0  1
For example, row n = 9 counts the following partitions:
  (7,1,1)              (5,2,2)      (3,3,3)  (4,4,1)  .  .  .  .  (9)
  (3,3,1,1,1)          (6,2,1)      (4,3,2)
  (4,2,1,1,1)          (2,2,2,2,1)  (5,3,1)
  (5,1,1,1,1)          (3,2,2,1,1)
  (2,2,1,1,1,1,1)
  (3,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1)

Crossrefs

Column k=1 is A002865(n-1).
Row sums are A027193 (odd-length ptns), strict A067659.
This is the odd-length case of A359901, with half-steps A359893.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
A325347 counts partitions w/ integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
Cf. A008289, A026424, A327472, A359889, A359895, A359906, A359907, A359910.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Median[#]==k&]],{n,15},{k,n}]

A359894 Number of integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 10, 13, 20, 28, 49, 53, 93, 113, 145, 203, 287, 329, 479, 556, 724, 955, 1242, 1432, 1889, 2370, 2863, 3502, 4549, 5237, 6825, 8108, 9839, 12188, 14374, 16958, 21617, 25852, 30582, 36100, 44561, 51462, 63238, 73386, 85990, 105272, 124729
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Examples

			The a(4) = 1 through a(8) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)
         (311)   (3111)   (331)     (422)
         (2111)  (21111)  (421)     (431)
                          (511)     (521)
                          (2221)    (611)
                          (3211)    (4211)
                          (4111)    (5111)
                          (22111)   (22211)
                          (31111)   (32111)
                          (211111)  (41111)
                                    (221111)
                                    (311111)
                                    (2111111)

Crossrefs

The complement is counted by A240219.
These partitions are ranked by A359890, complement A359889.
The odd-length case is ranked by A359892, complement A359891.
The odd-length case is A359896, complement A359895.
The strict case is A359898, complement A359897.
The odd-length strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284 and A058398 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.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
A359909 counts factorizations with the same mean as median, odd-len A359910.
Cf. A066571, A316313, A327472, A327482, A349156, A359906.

Programs

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

A360009 Numbers whose prime indices have integer mean and integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110, 111
Offset: 1

Views

Author

Gus Wiseman, Jan 24 2023

Keywords

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 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 terms together with their prime indices begin:
    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}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}

Crossrefs

For just integer mean we have A316413 (counted by A067538).
The mean of prime indices is given by A326567/A326568.
The complement is A348551 \/ A359912 (counted by A349156 and A307683).
These partitions are counted by A359906.
For just integer median we have A359908 (counted by A325347).
The median of prime indices is given by A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A327473, A359889, A359890, A359903, A359905, A360006.

Programs

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

Formula

Intersection of A316413 and A359908.

A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 1, 5, 5, 2, 5, 2, 8, 18, 1, 2, 19, 2, 24, 41, 20, 2, 9, 44, 31, 94, 102, 2, 125, 2, 1, 206, 68, 365, 382, 2, 98, 433, 155, 2, 716, 2, 1162, 2332, 196, 2, 17, 1108, 563, 1665, 3287, 2, 3906, 5474, 2005, 3083, 509, 2, 9029
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Comments

The length and median of such a partition are integers with product n.

Examples

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)  (9)
            (111)       (11111)  (222)  (1111111)       (333)
                                 (321)                  (432)
                                                        (531)
                                                        (111111111)
The a(15) = 18 partitions:
  (15)
  (5,5,5)
  (6,5,4)
  (7,5,3)
  (8,5,2)
  (9,5,1)
  (3,3,3,3,3)
  (4,3,3,3,2)
  (4,4,3,2,2)
  (4,4,3,3,1)
  (5,3,3,2,2)
  (5,3,3,3,1)
  (5,4,3,2,1)
  (5,5,3,1,1)
  (6,3,3,2,1)
  (6,4,3,1,1)
  (7,3,3,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

Links

Crossrefs

This is the odd-length case of A240219, complement A359894, strict A359897.
These partitions are ranked by A359891, complement A359892.
The complement is counted by A359896.
The strict case is A359899, complement A359900.
The version for factorizations is A359910.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
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. A008289, A316313, A327472, A327475, A327482, A359889, A359906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]
  • PARI
    \\ P(n, k, m) is g.f. for k parts of max size m.
P(n, k, m)={polcoef(1/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)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ Andrew Howroyd, Jan 21 2023

Formula

a(p) = 2 for prime p. - Andrew Howroyd, Jan 21 2023

A359910 Number of odd-length integer factorizations of n into factors > 1 with the same mean as median.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3
Offset: 1

Views

Author

Gus Wiseman, Jan 24 2023

Keywords

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(n) factorizations for n = 120, 960, 5760, 6720:
  120      960         5760            6720
  4*5*6    2*16*30     16*18*20        4*30*56
  2*6*10   4*12*20     3*5*6*8*8       10*21*32
           8*10*12     4*4*6*6*10      12*20*28
           3*4*4*4*5   2*2*8*10*18     4*5*6*7*8
                       2*2*2*4*4*5*9   2*4*7*10*12
                                       2*2*2*4*5*6*7

Links

Crossrefs

The version for partitions is A359895, ranked by A359891.
This is the odd-length case of A359909, partitions A240219.
A001055 counts factorizations.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A316313, A326567/A326568, A359889, A359894, A359897, A359902, A359906, A359911, A360005.

Programs

  • 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],OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,100}]
  • PARI
    A359910(n, m=n, facs=List([])) = if(1==n, (((#facs)%2) && (facs[(1+#facs)/2]==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359910(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Extensions

More terms from Antti Karttunen, Jan 20 2025

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

Author

Gus Wiseman, Jan 20 2023

Keywords

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

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

A359900 Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 4, 8, 10, 8, 15, 18, 17, 26, 27, 31, 43, 51, 53, 59, 81, 87, 109, 127, 115, 169, 194, 213, 255, 243, 322, 379, 431, 478, 487, 629, 667, 804, 907, 902, 1151, 1294, 1439, 1530, 1674, 2031, 2290, 2559, 2829, 2973, 3296, 3939
Offset: 0

Views

Author

Gus Wiseman, Jan 21 2023

Keywords

Examples

			The a(7) = 1 through a(16) = 15 partitions (A=10, B=11, C=12, D=13):
  (421)  (431)  (621)  (532)  (542)  (651)  (643)  (653)  (762)  (754)
         (521)         (541)  (632)  (732)  (652)  (743)  (843)  (763)
                       (631)  (641)  (831)  (742)  (752)  (861)  (853)
                       (721)  (731)  (921)  (751)  (761)  (942)  (862)
                              (821)         (832)  (842)  (A32)  (871)
                                            (841)  (851)  (A41)  (943)
                                            (931)  (932)  (B31)  (952)
                                            (A21)  (941)  (C21)  (961)
                                                   (A31)         (A42)
                                                   (B21)         (A51)
                                                                 (B32)
                                                                 (B41)
                                                                 (C31)
                                                                 (D21)
                                                                 (64321)

Crossrefs

This is the strict case of A359896, complement A359895, ranked by A359892.
This is the odd-length case of A359898, complement A359897.
The complement is counted by A359899.
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.
A359893/A359901/A359902 count partitions by median, ranked by A360005.
Cf. A000016, A065795, A066571, A102627, A240850, A240851, A327475, A359894, A359906, A359907, A359910.

Programs

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

A359909 Number of integer factorizations of n into factors > 1 with the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 4, 2, 2, 2, 6, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 7, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 6, 4, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 4, 1, 4, 5, 2, 1, 6, 1, 4, 2, 5, 1, 4, 2, 3, 3, 2, 2, 11
Offset: 1

Views

Author

Gus Wiseman, Jan 24 2023

Keywords

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(n) factorizations for n = 24, 36, 60, 120, 144, 360:
  24      36        60      120       144       360
  3*8     4*9       2*30    2*60      2*72      4*90
  4*6     6*6       3*20    3*40      3*48      5*72
  2*12    2*18      4*15    4*30      4*36      6*60
  2*3*4   3*12      5*12    5*24      6*24      8*45
          2*2*3*3   6*10    6*20      8*18      9*40
                    3*4*5   8*15      9*16      10*36
                            10*12     12*12     12*30
                            4*5*6     2*2*6*6   15*24
                            2*6*10    3*3*4*4   18*20
                            2*3*4*5             2*180
                                                3*120
                                                2*10*18
                                                3*4*5*6

Links

Crossrefs

The version for partitions is A240219, complement A359894.
These multisets are ranked by A359889.
The version for strict partitions is A359897.
The odd-length case is A359910.
The complement is counted by A359911.
A001055 counts factorizations.
A058398 counts partitions by mean, see also A008284, A327482.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A316313, A326567/A326568, A359906, A360005.

Programs

  • 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],Mean[#]==Median[#]&]],{n,100}]
  • PARI
    median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
A359909(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359909(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Extensions

More terms from Antti Karttunen, Jan 20 2025

A359896 Number of odd-length integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 6, 9, 11, 15, 27, 32, 50, 58, 72, 112, 149, 171, 246, 286, 359, 477, 630, 773, 941, 1181, 1418, 1749, 2289, 2668, 3429, 4162, 4878, 6074, 7091, 8590, 10834, 12891, 15180, 18491, 22314, 25845, 31657, 36394, 42269, 52547, 62414, 73576, 85701
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Examples

			The a(4) = 1 through a(9) = 11 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (21111)  (331)    (422)      (522)
                         (421)    (431)      (621)
                         (511)    (521)      (711)
                         (22111)  (611)      (22221)
                         (31111)  (22211)    (32211)
                                  (32111)    (33111)
                                  (41111)    (42111)
                                  (2111111)  (51111)
                                             (2211111)
                                             (3111111)

Crossrefs

These partitions are ranked by A359892.
The any-length version is A359894, complement A240219, strict A359898.
The complement is counted by A359895, ranked by A359891.
The strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
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. A008289, A066571, A316313, A327472, A327482, A359890, A359906, A359910.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]!=Median[#]&]],{n,0,30}]
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