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 10 results.

A086543 Number of partitions of n with at least one odd part.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 15, 17, 30, 35, 56, 66, 101, 120, 176, 209, 297, 355, 490, 585, 792, 946, 1255, 1498, 1958, 2335, 3010, 3583, 4565, 5428, 6842, 8118, 10143, 12013, 14883, 17592, 21637, 25525, 31185, 36711, 44583, 52382, 63261, 74173, 89134, 104303, 124754, 145698, 173525, 202268
Offset: 0

Views

Author

Vladeta Jovovic, Sep 10 2003

Keywords

Comments

From Gus Wiseman, Oct 12 2023: (Start)
Also the number of integer partitions of n whose greatest part is not n/2, ranked by A366319. The a(1) = 1 through a(7) = 15 partitions are:
(1) (2) (3) (4) (5) (6) (7)
(21) (31) (32) (42) (43)
(111) (1111) (41) (51) (52)
(221) (222) (61)
(311) (411) (322)
(2111) (2211) (331)
(11111) (21111) (421)
(111111) (511)
(2221)
(3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
Compare to the a(1) = 1 through a(7) = 15 partitions with at least one odd part, ranked by A366322:
(1) (11) (3) (31) (5) (33) (7)
(21) (211) (32) (51) (43)
(111) (1111) (41) (321) (52)
(221) (411) (61)
(311) (2211) (322)
(2111) (3111) (331)
(11111) (21111) (421)
(111111) (511)
(2221)
(3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
(End)

Examples

			a(4)=3 because we have [3,1],[2,1,1] and [1,1,1] ([4] and [2,2] do not qualify).

Links

Crossrefs

Cf. A038348, A047967.
The complement is counted by A035363, ranks A344415.
These partitions have ranks A366322.
A025065 counts partitions with sum <= twice length, ranks A344296.
A110618 counts partitions with sum >= twice maximum, ranks A344291.
Cf. A238628, A257991, A300061, A320924, A340387, A344414, A366319.

Programs

  • Maple
    g:=sum(x^(2*k-1)/product(1-x^j,j=1..2*k-1)/product(1-x^(2*j),j=k..70),k=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..45); # Emeric Deutsch, Mar 30 2006
  • Mathematica
    nn=50;CoefficientList[Series[Sum[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{k,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 28 2013 *)
Table[Length[Select[IntegerPartitions[n],Max[#]!=n/2&]],{n,0,30}] (* Gus Wiseman, Oct 12 2023 *)
  • PARI
    x='x+O('x^66); concat([0], Vec(1/eta(x)-1/eta(x^2)) ) \\ Joerg Arndt, May 04 2013

Formula

A000041(n) if n is odd; otherwise, A000041(n) - A000041(n/2).
G.f.: Sum_{k>=1} x^(2k-1)/((Product_{j=1..2k-1} (1-x^j))*(Product_{j>=k} (1-x^(2j)))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/E(x) - 1/E(x^2) where E(x) = prod(n>=1, 1-x^n ); see Pari code. - Joerg Arndt, May 04 2013

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2
Offset: 1

Views

Author

Gus Wiseman, Oct 22 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, sum A056239(n).

Examples

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Crossrefs

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Programs

  • Mathematica
    Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

Formula

a(n) = A056239(n) - A366531(n).

A182616 Number of partitions of 2n that contain odd parts.

Original entry on oeis.org

0, 1, 3, 8, 17, 35, 66, 120, 209, 355, 585, 946, 1498, 2335, 3583, 5428, 8118, 12013, 17592, 25525, 36711, 52382, 74173, 104303, 145698, 202268, 279153, 383145, 523105, 710655, 960863, 1293314, 1733281, 2313377, 3075425, 4073085, 5374806, 7067863, 9263076
Offset: 0

Views

Author

Omar E. Pol, Dec 03 2010

Keywords

Comments

Bisection (even part) of A086543.

Examples

			For n=3 the partitions of 2n are
6 ....................... does not contains odd parts
3 + 3 ................... contains odd parts ........... *
4 + 2 ................... does not contains odd parts
2 + 2 + 2 ............... does not contains odd parts
5 + 1 ................... contains odd parts ........... *
3 + 2 + 1 ............... contains odd parts ........... *
4 + 1 + 1 ............... contains odd parts ........... *
2 + 2 + 1 + 1 ........... contains odd parts ........... *
3 + 1 + 1 + 1 ........... contains odd parts ........... *
2 + 1 + 1 + 1 + 1 ....... contains odd parts ........... *
1 + 1 + 1 + 1 + 1 + 1 ... contains odd parts ........... *
There are 8 partitions of 2n that contain odd parts.
Also p(2n)-p(n) = p(6)-p(3) = 11-3 = 8, where p(n) is the number of partitions of n, so a(3)=8.
From _Gus Wiseman_, Oct 18 2023: (Start)
For n > 0, also the number of integer partitions of 2n that do not contain n, ranked by A366321. For example, the a(1) = 1 through a(4) = 17 partitions are:
  (2)  (4)     (6)       (8)
       (31)    (42)      (53)
       (1111)  (51)      (62)
               (222)     (71)
               (411)     (332)
               (2211)    (521)
               (21111)   (611)
               (111111)  (2222)
                         (3221)
                         (3311)
                         (5111)
                         (22211)
                         (32111)
                         (221111)
                         (311111)
                         (2111111)
                         (11111111)
(End)

Crossrefs

Cf. A304710.
Bisection of A086543, with ranks A366322.
The case of all odd parts is A035294, bisection of A000009.
The strict case is A365828.
These partitions have ranks A366530.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions with at least one odd and even part, ranks A366532.
A047967 counts partitions with at least one even part, ranks A324929.
A086543 counts partitions of n not containing n/2, ranks A366319.
A366527 counts partitions of 2n with an even part, ranks A366529.
Cf. A001522, A006827, A058695, A078408, A079122, A231429, A365543, A366321.

Programs

  • Maple
    with(combinat): a:= n-> numbpart(2*n) -numbpart(n): seq(a(n), n=0..35);
  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],n>0&&FreeQ[#,n]&]],{n,0,15}] (* Gus Wiseman, Oct 11 2023 *)
Table[Length[Select[IntegerPartitions[2n],Or@@OddQ/@#&]],{n,0,15}] (* Gus Wiseman, Oct 11 2023 *)

Formula

a(n) = A000041(2*n) - A000041(n).

Extensions

Edited by Alois P. Heinz, Dec 03 2010

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 3, 4, 3, 5, 5, 3, 4, 4, 6, 5, 6, 6, 5, 8, 7, 5, 6, 8, 6, 10, 7, 10, 9, 10, 8, 12, 11, 7, 10, 12, 12, 10, 15, 11, 14, 15, 15, 16, 12, 18, 15, 11, 14, 20, 18, 20, 15, 22, 15, 22, 21, 25, 24, 24, 18, 27
Offset: 0

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

The version for all k = 0..n is A113685 (including zeros).

Examples

			Triangle begins:
   1
   1
   1  1
   1  2
   2  1  2
   2  2  3
   3  2  2  4
   3  4  3  5
   5  3  4  4  6
   5  6  6  5  8
   7  5  6  8  6 10
   7 10  9 10  8 12
  11  7 10 12 12 10 15
  11 14 15 15 16 12 18
  15 11 14 20 18 20 15 22
  15 22 21 25 24 24 18 27
Row n = 8 counts the following partitions:
  (8)     (611)    (431)     (521)      (71)
  (62)    (4211)   (41111)   (332)      (53)
  (44)    (22211)  (3221)    (32111)    (5111)
  (422)            (221111)  (2111111)  (3311)
  (2222)                                (311111)
                                        (11111111)
Row n = 9 counts the following partitions:
  (81)     (63)      (54)       (72)        (9)
  (621)    (6111)    (522)      (5211)      (711)
  (441)    (432)     (4311)     (3321)      (531)
  (4221)   (42111)   (411111)   (321111)    (51111)
  (22221)  (3222)    (32211)    (21111111)  (333)
           (222111)  (2211111)              (33111)
                                            (3111111)
                                            (111111111)

Crossrefs

Row sums are A000041.
The version including all k is A113685, even version A113686.
Column k = 1 is A119620.
The even version and the reverse version are both A174713.
For odd-indexed instead of odd parts we have A346697, even version A346698.
The corresponding rank statistic is A366528, even version A366531.
A000009 counts partitions into odd parts, ranks A066208.
A086543 counts partitions with odd parts, ranks A366322.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A045931, A053253, A066967, A130780, A171966, A241638, A268335, A325698, A366533.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Total[Select[#,OddQ]]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

Formula

T(n,k) = A000009(k) * A000041((n-k)/2).

A366531 Sum of even prime indices of n.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 4, 0, 4, 0, 0, 2, 6, 4, 2, 0, 0, 4, 8, 0, 6, 0, 0, 2, 0, 6, 6, 4, 10, 2, 0, 0, 2, 0, 4, 4, 12, 8, 8, 0, 0, 6, 14, 0, 4, 0, 0, 2, 8, 0, 2, 6, 16, 6, 0, 4, 10, 10, 0, 2, 18, 0, 8, 0, 6, 2, 0, 0, 2, 4, 20, 4, 0, 12, 2, 8, 4, 8, 22, 0, 8, 0, 0, 6
Offset: 1

Views

Author

Gus Wiseman, Oct 22 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.

Examples

			The prime indices of 198 are {1,2,2,5}, so a(198) = 2+2 = 4.

Crossrefs

Zeros are A066208, counted by A000009.
The triangle for the odd version is A113685, without zeros A365067.
The triangle for this statistic is A113686, without zeros A174713.
The odd version is A366528.
The halved version is A366533.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A239261 counts partitions with sum of odd parts = sum of even parts.
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Programs

  • Mathematica
    Table[Total[Cases[FactorInteger[n], {p_?(EvenQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

Formula

a(n) = A056239(n) - A366528(n).

A366533 Sum of even prime indices of n divided by 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 3, 2, 1, 0, 0, 2, 4, 0, 3, 0, 0, 1, 0, 3, 3, 2, 5, 1, 0, 0, 1, 0, 2, 2, 6, 4, 4, 0, 0, 3, 7, 0, 2, 0, 0, 1, 4, 0, 1, 3, 8, 3, 0, 2, 5, 5, 0, 1, 9, 0, 4, 0, 3, 1, 0, 0, 1, 2, 10, 2, 0, 6, 1, 4, 2, 4, 11, 0, 4, 0, 0, 3, 0, 7
Offset: 1

Views

Author

Gus Wiseman, Oct 23 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.

Examples

			The prime indices of 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.

Links

Crossrefs

Zeros are A066208, counted by A000009.
The triangle for this statistic (without zeros) is A174713.
The un-halved odd version is A366528.
The un-halved version is A366531.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A365067 counts partitions by sum of odd parts (without zeros).
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Programs

  • Maple
    f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023
  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

Formula

a(n) = A366531(n)/2.

A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

Original entry on oeis.org

1, 3, 7, 10, 13, 16, 19, 21, 22, 27, 28, 29, 34, 36, 37, 39, 43, 46, 48, 52, 53, 55, 57, 61, 62, 64, 66, 71, 75, 76, 79, 81, 82, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 113, 115, 116, 117, 118, 120, 129, 130, 131, 133, 134, 136, 138, 139, 144
Offset: 0

Views

Author

Gus Wiseman, Oct 13 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.

Examples

			The prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.
The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   10: {1,3}
   13: {6}
   16: {1,1,1,1}
   19: {8}
   21: {2,4}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   34: {1,7}
   36: {1,1,2,2}

Crossrefs

Partitions of this type are counted by A182616, strict A365828.
A066207 lists numbers with all even prime indices, odd A066208.
A086543 lists numbers with at least one odd prime index, counted by A366322.
A300063 ranks partitions of odd numbers.
A366319 ranks partitions of n not containing n/2.
A366321 ranks partitions of 2k that do not contain k.
Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[prix[#]]]&&FreeQ[prix[#],Total[prix[#]]/2]&]

A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

Original entry on oeis.org

4, 10, 12, 16, 22, 25, 28, 30, 34, 36, 40, 46, 48, 52, 55, 62, 64, 66, 70, 75, 76, 82, 84, 85, 88, 90, 94, 100, 102, 108, 112, 115, 116, 118, 120, 121, 130, 134, 136, 138, 144, 146, 148, 154, 155, 156, 160, 165, 166, 172, 175, 184, 186, 187, 190, 192, 194, 196
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices are the following. Each multiset has even sum and at least one odd part.
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   46: {1,9}
   48: {1,1,1,1,2}
   52: {1,1,6}
   55: {3,5}
   62: {1,11}
   64: {1,1,1,1,1,1}

Crossrefs

These partitions are counted by A182616, even bisection of A086543.
Not requiring at least one odd part gives A300061.
Allowing partitions of odd numbers gives A366322.
A031368 lists primes of odd index.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A047967, A257992, A324929, A358137.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], EvenQ[Total[prix[#]]]&&Or@@OddQ/@prix[#]&]

A366527 Number of integer partitions of 2n containing at least one even part.

Original entry on oeis.org

0, 1, 3, 7, 16, 32, 62, 113, 199, 339, 563, 913, 1453, 2271, 3496, 5308, 7959, 11798, 17309, 25151, 36225, 51748, 73359, 103254, 144363, 200568, 277007, 380437, 519715, 706412, 955587, 1286762, 1725186, 2303388, 3063159, 4058041, 5356431, 7045454, 9235841
Offset: 0

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

Also partitions of 2n with even product.

Examples

			The a(1) = 1 through a(4) = 16 partitions:
  (2)  (4)    (6)      (8)
       (22)   (42)     (44)
       (211)  (222)    (62)
              (321)    (332)
              (411)    (422)
              (2211)   (431)
              (21111)  (521)
                       (611)
                       (2222)
                       (3221)
                       (4211)
                       (22211)
                       (32111)
                       (41111)
                       (221111)
                       (2111111)

Crossrefs

This is the even bisection of A047967.
For odd instead of even parts we have A182616, ranks A366321 or A366528.
These partitions have ranks A366529, subset of A324929.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions w/ at least one odd and even part, ranks A366532.
A086543 counts partitions of n not containing n/2, ranks A366319.
A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]

Formula

a(n) = A000041(2n) - A000009(2n).

A366532 Heinz numbers of integer partitions with at least one even and odd part.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 30, 33, 35, 36, 38, 42, 45, 48, 51, 52, 54, 56, 58, 60, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 90, 93, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 114, 116, 119, 120, 122, 123, 126, 130, 132, 135, 138, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

These partitions are counted by A006477.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Crossrefs

These partitions are counted by A006477.
Just even: A324929, counted by A047967.
Just odd: A366322, counted by A086543 (even bisection of A182616).
A031368 lists primes of odd index, even A031215.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 lists prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
A257992 counts even prime indices, distinct A324967.
Cf. A000720, A001222, A003963, A026804, A244991, A257992.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or@@EvenQ/@prix[#]&&Or@@OddQ/@prix[#]&]

Formula

Intersection of A324929 and A366322.
Showing 1-10 of 10 results.

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