A295342 -id:A295342 - cp's OEIS frontend

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

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).

A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 3, 1, 1, 7, 4, 2, 1, 1, 11, 7, 4, 2, 1, 1, 15, 10, 6, 3, 2, 1, 1, 22, 16, 9, 6, 3, 2, 1, 1, 30, 22, 14, 8, 5, 3, 2, 1, 1, 42, 32, 20, 13, 8, 5, 3, 2, 1, 1, 56, 44, 29, 18, 12, 7, 5, 3, 2, 1, 1, 77, 62, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Gus Wiseman, May 22 2022

Also partitions of n with at least one part appearing k or more times. It would be interesting to have a bijective proof of this.

			Triangle begins:
   1
   2  1
   3  1  1
   5  3  1  1
   7  4  2  1  1
  11  7  4  2  1  1
  15 10  6  3  2  1  1
  22 16  9  6  3  2  1  1
  30 22 14  8  5  3  2  1  1
  42 32 20 13  8  5  3  2  1  1
  56 44 29 18 12  7  5  3  2  1  1
  77 62 41 27 17 12  7  5  3  2  1  1
For example, row n = 5 counts the following partitions:
  (5)      (32)    (32)   (41)  (5)
  (32)     (41)    (311)
  (41)     (221)
  (221)    (2111)
  (311)
  (2111)
  (11111)
At least one part appearing k or more times:
  (5)      (221)    (2111)   (11111)  (11111)
  (32)     (311)    (11111)
  (41)     (2111)
  (221)    (11111)
  (311)
  (2111)
  (11111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

The complement is counted by A061199.
Differences of consecutive terms are A091602.
Column k = 1 is A000041.
Column k = 2 is A047967, ranked by A013929 and A324929.
Column k = 3 is A295341, ranked by A046099 and A354235.
Column k = 4 is A295342.
A000041 counts integer partitions, strict A000009.
A047966 counts uniform partitions.
Cf. A002033, A006918, A064410, A117485, A238394, A238395, A325534.

Table[Length[Select[IntegerPartitions[n],MemberQ[#/k,_?IntegerQ]&]],{n,1,15},{k,1,n}]
- or -
Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]>=k&]],{n,1,15},{k,1,n}]

\\ here P(k,n) is partitions with no part divisible by k as g.f.
P(k,n)={1/prod(i=1, n, 1 - if(i%k, x^i) + O(x*x^n))}
M(n,m=n)={my(p=P(n+1,n)); Mat(vector(m, k, Col(p-P(k,n), -n) ))}
{ my(A=M(12)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Jan 19 2023

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Original entry on oeis.org

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

Offset: 1

Gus Wiseman, May 23 2022

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:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}

For 4 instead of 3 we have A046101, counted by A295342.
This sequence ranks the partitions counted by A295341, compositions A335464.
For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A004709 lists numbers divisible by no cube, counted by A000726.
A036966 lists 3-full numbers, counted by A100405.
A046099 lists non-cubefree numbers.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A354234 counts partitions of n with at least one part divisible by k.
Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]

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

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A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

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A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

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