A324929 -id:A324929 - cp's OEIS frontend

A047967 Number of partitions of n with some part repeated.

0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 32, 44, 62, 83, 113, 149, 199, 259, 339, 436, 563, 716, 913, 1151, 1453, 1816, 2271, 2818, 3496, 4309, 5308, 6502, 7959, 9695, 11798, 14298, 17309, 20877, 25151, 30203, 36225, 43323, 51748, 61651, 73359, 87086, 103254, 122164

Offset: 0

N. J. A. Sloane

nonn

Also number of partitions of n with at least one even part. - Vladeta Jovovic, Sep 10 2003. Example: a(5)=4 because we have [4,1], [3,2], [2,2,1] and [2,1,1,1] ([5], [3,1,1] and [1,1,1,1,1] do not qualify). - Emeric Deutsch, Mar 30 2006
Also number of partitions of n (where it is assumed that the least part is 0) such that at least one difference is at least two. Example: a(5)=4 because we have [5,0], [4,1,0], [3,2,0] and [3,1,1,0] ([2,2,1,0], [2,1,1,1,0] and [1,1,1,1,1,0] do not qualify). - Emeric Deutsch, Mar 30 2006
The Heinz numbers of these partitions (with some part repeated) are given by A013929. Equivalent to Vladeta Jovovic's comment, a(n) is also the number of integer partitions whose product of parts is even. The Heinz numbers of these latter partitions are given by A324929. - Gus Wiseman, Mar 23 2019

			a(5) = 4 because we have [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([5], [4,1] and [3,2] do not qualify).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
H. Bottomley, Illustration for A000009, A000041, A047967.

Cf. A038348, A261982.
Column k=1 of A320264.
Cf. A324847, A324929, A324966, A324967.

g:=sum(x^(2*k)*product(1+x^j,j=k+1..70)/product(1-x^j,j=1..k),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..44); # Emeric Deutsch, Mar 30 2006

Table[PartitionsP[n]-PartitionsQ[n],{n,0,50}] (* Harvey P. Dale, Jan 17 2019 *)

x='x+O('x^66); concat([0,0], Vec(1/eta(x)-eta(x^2)/eta(x))) \\ Joerg Arndt, Jun 21 2011

a(n) = A000041(n) - A000009(n).
G.f.: Sum_{k>=1} x^(2*k)*(Product_{j>=k+1} (1+x^j)) / Product_{j=1..k} (1-x^j) = Sum_{k>=1} x^(2*k)/(Product_{j=1..2*k} (1-x^j)*Product_{j>=k} (1-x^(2*j+1))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/P(x) - P(x^2)/P(x) where P(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Jun 21 2011
a(n) = p(n-2)+p(n-4)-p(n-10)-p(n-14)+...+(-1^(j-1))*p(n-j*(3*j-1)) + (-1^(j-1))*p(n-j*(3*j+1))+..., where p(n) = A000041(n). - Gregory L. Simay, Aug 28 2023

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

Omar E. Pol, Dec 03 2010

nonn

Bisection (even part) of A086543.

			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)

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.

with(combinat): a:= n-> numbpart(2*n) -numbpart(n): seq(a(n), n=0..35);

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 *)

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

Edited by Alois P. Heinz, Dec 03 2010

A324966 Number of distinct odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 0

Offset: 1

Gus Wiseman, Mar 21 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.

If x and y are coprime then a(x*y) = a(x)+a(y). - Robert Israel, Mar 24 2019

			180180 has prime indices {1,1,2,2,3,4,5,6}, so a(180180) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A003963, A005087, A066208, A112798, A257991, A257992, A289509, A324929, A324967.

f:= proc(n) nops(select(type,map(numtheory:-pi,numtheory:-factorset(n)),odd)) end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019

Table[Count[If[n==1,{},FactorInteger[n]],{?(OddQ[PrimePi[#]]&),}],{n,100}]

a(n) = my(f=factor(n)[,1]); sum(k=1, #f, primepi(f[k]) % 2); \\ Michel Marcus, Mar 22 2019

a(n) = A001221(n) - A324967(n). - Robert Israel, Mar 24 2019
G.f.: Sum_{k>=1} x^prime(2*k-1) / (1 - x^prime(2*k-1)). - Ilya Gutkovskiy, Feb 12 2020
Additive with a(p^e) = 1 if primepi(p) is odd and 0 otherwise. - Amiram Eldar, Oct 06 2023

A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Oct 14 2023

nonn

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.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    5: {3}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}

The complement is A066207, counted by A035363.
For all odd parts we have A066208, counted by A000009.
Partitions of this type are counted by A086543.
For even instead of odd we have A324929, counted by A047967.
A031368 lists primes of odd index.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A257992, A318400, A324927, A358137.

Select[Range[100],Or@@OddQ/@PrimePi/@First/@FactorInteger[#]&]

A257991(a(n)) > 0.

A324856 Numbers divisible by exactly one of their prime indices.

Original entry on oeis.org

2, 10, 14, 15, 22, 26, 34, 38, 45, 46, 50, 55, 58, 62, 70, 74, 82, 86, 94, 98, 105, 106, 118, 119, 122, 130, 134, 135, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 195, 202, 206, 207, 214, 218, 226, 230, 242, 250, 254, 255, 262, 266, 274, 275, 278, 285

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A324848(n) = 1.
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.
If k is in A324846, then k*prime(k) is in the sequence. - Robert Israel, Mar 22 2019

			The sequence of terms together with their prime indices begins:
   2: {1}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  22: {1,5}
  26: {1,6}
  34: {1,7}
  38: {1,8}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  55: {3,5}
  58: {1,10}
  62: {1,11}
  70: {1,3,4}
  74: {1,12}
  82: {1,13}
  86: {1,14}
  94: {1,15}
  98: {1,4,4}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A112798, A120383, A323440, A324694, A324704, A324846, A324847, A324848, A324849, A324850, A324926, A324929.

filter:= proc(n) local F;
  F:= select(t -> n mod numtheory:-pi(t[1])=0, ifactors(n)[2]);
  nops(F)=1 and F[1][2]=1
end proc:
select(filter, [$2..1000]); # Robert Israel, Mar 22 2019

Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k/;Divisible[#,PrimePi[p]]]]==1&]

A324967 Number of distinct even prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 0, 0, 2, 0, 1, 2

Offset: 1

Gus Wiseman, Mar 21 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.

If x and y are coprime then a(x*y) = a(x) + a(y). - Robert Israel, Mar 24 2019

			180180 has prime indices {1,1,2,2,3,4,5,6}, so a(180180) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A003963, A005087, A066208, A112798, A257991, A257992, A289509, A324929, A324966.

f:= proc(n) nops(select(type,map(numtheory:-pi,numtheory:-factorset(n)),even)) end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019

Table[Count[If[n==1,{},FactorInteger[n]],{?(EvenQ[PrimePi[#]]&),}],{n,100}]

a(n) = my(f=factor(n)[,1]); sum(k=1, #f, !(primepi(f[k]) % 2)); \\ Michel Marcus, Mar 22 2019

a(n) = A001221(n) - A324966(n). - Robert Israel, Mar 24 2019
G.f.: Sum_{k>=1} x^prime(2*k) / (1 - x^prime(2*k)). - Ilya Gutkovskiy, Feb 12 2020
Additive with a(p^e) = 1 if primepi(p) is even and 0 otherwise. - Amiram Eldar, Oct 06 2023

A323440 Numbers divisible by exactly one of their distinct prime indices.

Original entry on oeis.org

2, 4, 8, 10, 14, 15, 16, 20, 22, 26, 32, 34, 38, 40, 44, 45, 46, 50, 52, 55, 58, 62, 64, 68, 70, 74, 75, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 105, 106, 116, 118, 119, 122, 124, 128, 130, 134, 135, 136, 142, 146, 148, 154, 158, 160, 164, 166, 170, 172, 176

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A324852(n) = 1.

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 sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   8: {1,1,1}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  20: {1,1,3}
  22: {1,5}
  26: {1,6}
  32: {1,1,1,1,1}
  34: {1,7}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  52: {1,1,6}
  55: {3,5}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A112798, A120383, A324704, A324846, A324847, A324848, A324849, A324850, A324856, A324926, A324929.

Select[Range[100],Count[If[#==1,{},FactorInteger[#]],{p_,_}/;Divisible[#,PrimePi[p]]]==1&]

isok(n) = my(f=factor(n)[,1]); sum(k=1, #f, (n % primepi(f[k])) == 0) == 1; \\ Michel Marcus, Mar 22 2019

A324927 Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 18, 19, 21, 24, 27, 28, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 262, 266, 288, 294, 304, 311, 318

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A109082(n) = 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.
Also Heinz numbers of integer partitions into powers of 2 with at least one part > 1 (counted by A102378).

			The sequence of terms together with their prime indices begins:
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  14: {1,4}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  24: {1,1,1,2}
  27: {2,2,2}
  28: {1,1,4}
  36: {1,1,2,2}
  38: {1,8}
  42: {1,2,4}
  48: {1,1,1,1,2}
  49: {4,4}
  53: {16}
  54: {1,2,2,2}
  56: {1,1,1,4}

Cf. A000081, A000720, A003963, A007097, A018819, A033844, A056239, A102378, A112798, A302242, A318400, A324928, A324929.

Select[Range[100],And[!IntegerQ[Log[2,#]],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Log[2,PrimePi[p]]]]]&]

A356945 Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 08 2022

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 a{n} multiset partitions for n = 8, 24, 72, 96:
  {{111}}      {{1112}}      {{11122}}      {{111112}}
  {{1}{11}}    {{1}{112}}    {{1}{1122}}    {{1}{11112}}
  {{1}{1}{1}}  {{11}{12}}    {{11}{122}}    {{11}{1112}}
               {{1}{1}{12}}  {{12}{112}}    {{111}{112}}
                             {{1}{1}{122}}  {{12}{1111}}
                             {{1}{12}{12}}  {{1}{1}{1112}}
                                            {{1}{11}{112}}
                                            {{11}{11}{12}}
                                            {{1}{12}{111}}
                                            {{1}{1}{1}{112}}
                                            {{1}{1}{11}{12}}
                                            {{1}{1}{1}{1}{12}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Positions of 0's are A080259, complement A055932.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Multisets covering an initial interval are counted by A000009, A000041, A011782, ranked by A055932.
Other types: A034691, A089259, A356954, A356955.
Other conditions: A050320, A050330, A322585, A356233, A356931, A356936.
Cf. A003963, A073491, A107742, A287170, A324929, A340852, A356234.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nnQ[m_]:=PrimePi/@First/@FactorInteger[m]==Range[PrimePi[Max@@First/@FactorInteger[m]]];
Table[Length[Select[facs[n],And@@nnQ/@#&]],{n,100}]

A295341 The number of partitions of n in which at least one part is a multiple of 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 6, 9, 14, 20, 29, 41, 57, 78, 106, 142, 189, 250, 327, 425, 549, 705, 900, 1144, 1445, 1819, 2279, 2844, 3534, 4379, 5403, 6648, 8152, 9969, 12152, 14780, 17920, 21682, 26163, 31504, 37842, 45371, 54270, 64800, 77211, 91842, 109031, 129235, 152897

Offset: 0

R. J. Mathar, Nov 20 2017

nonn

From Gus Wiseman, May 23 2022: (Start)
Also the number of integer partitions of n with at least one part appearing more than twice. The Heinz numbers of these partitions are given by A046099. For example, the a(0) = 0 though a(8) = 9 partitions are:
  .  .  .  (111)  (1111)  (2111)   (222)     (2221)     (2222)
                          (11111)  (3111)    (4111)     (5111)
                                   (21111)   (22111)    (22211)
                                   (111111)  (31111)    (32111)
                                             (211111)   (41111)
                                             (1111111)  (221111)
                                                        (311111)
                                                        (2111111)
                                                        (11111111)
(End)

			From _Gus Wiseman_, May 23 2022: (Start)
The a(0) = 0 through a(8) = 9 partitions with a part that is a multiple of 3:
  .  .  .  (3)  (31)  (32)   (6)     (43)     (53)
                      (311)  (33)    (61)     (62)
                             (321)   (322)    (332)
                             (3111)  (331)    (431)
                                     (3211)   (611)
                                     (31111)  (3221)
                                              (3311)
                                              (32111)
                                              (311111)
(End)

The complement is counted by A000726, ranked by A004709.
These partitions are ranked by A354235.
This is column k = 3 of A354234.
For 2 instead of 3 we have A047967, ranked by A013929 and A324929.
For 4 instead of 3 we have A295342, ranked by A046101.
A000041 counts integer partitions, strict A000009.
A046099 lists non-cubefree numbers.
Cf. A001522, A006918, A064410, A064428, A117485, A188674, A325187, A325534.

Table[Length[Select[IntegerPartitions[n],MemberQ[#/3,?IntegerQ]&]],{n,0,30}] (* _Gus Wiseman, May 23 2022 *)
Table[Length[Select[IntegerPartitions[n],MatchQ[#,{_,x_,x_,x_,_}]&]],{n,0,30}] (* Gus Wiseman, May 23 2022 *)

a(n) = A000041(n)-A000726(n).

cp's OEIS Frontend

A047967 Number of partitions of n with some part repeated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A324966 Number of distinct odd prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A324856 Numbers divisible by exactly one of their prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A324967 Number of distinct even prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A323440 Numbers divisible by exactly one of their distinct prime indices.

Views

Author

Keywords

Comments

Examples

Links