A349157 -id:A349157 - cp's OEIS frontend

A349158 Heinz numbers of integer partitions with exactly one odd part.

2, 5, 6, 11, 14, 15, 17, 18, 23, 26, 31, 33, 35, 38, 41, 42, 45, 47, 51, 54, 58, 59, 65, 67, 69, 73, 74, 77, 78, 83, 86, 93, 95, 97, 98, 99, 103, 105, 106, 109, 114, 119, 122, 123, 126, 127, 135, 137, 141, 142, 143, 145, 149, 153, 157, 158, 161, 162, 167, 174

Offset: 1

Gus Wiseman, Nov 12 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with exactly one odd prime index. These are also partitions whose conjugate partition has alternating sum equal to 1.

Numbers that are product of a term of A031368 and a term of A066207. - Antti Karttunen, Nov 13 2021

			The terms and corresponding partitions begin:
      2: (1)         42: (4,2,1)       86: (14,1)
      5: (3)         45: (3,2,2)       93: (11,2)
      6: (2,1)       47: (15)          95: (8,3)
     11: (5)         51: (7,2)         97: (25)
     14: (4,1)       54: (2,2,2,1)     98: (4,4,1)
     15: (3,2)       58: (10,1)        99: (5,2,2)
     17: (7)         59: (17)         103: (27)
     18: (2,2,1)     65: (6,3)        105: (4,3,2)
     23: (9)         67: (19)         106: (16,1)
     26: (6,1)       69: (9,2)        109: (29)
     31: (11)        73: (21)         114: (8,2,1)
     33: (5,2)       74: (12,1)       119: (7,4)
     35: (4,3)       77: (5,4)        122: (18,1)
     38: (8,1)       78: (6,2,1)      123: (13,2)
     41: (13)        83: (23)         126: (4,2,2,1)

These partitions are counted by A000070 up to 0's.
Allowing no odd parts gives A066207, counted by A000041 up to 0's.
Requiring all odd parts gives A066208, counted by A000009.
These are the positions of 1's in A257991.
The even prime indices are counted by A257992.
The conjugate partitions are ranked by A345958.
Allowing at most one odd part gives A349150, counted by A100824.
A047993 ranks balanced partitions, counted by A106529.
A056239 adds up prime indices, row sums of A112798.
A122111 is a representation of partition conjugation.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A340604 ranks partitions of odd positive rank, counted by A101707.
A340932 ranks partitions whose least part is odd, counted by A026804.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000700, A001222, A027187, A027193, A028260, A031368 (primes with odd index), A035363, A215366, A277579, A300063, A349151.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],_?OddQ]==1&]

A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      13         15         18         20           22
   ------------------------------------------------------------------
    (2,1)  (6,5,2)    (10,5)     (12,6)     (12,7,1)     (12,8,2)
           (6,4,2,1)  (6,4,3,2)  (8,7,3)    (8,5,4,3)    (8,6,5,3)
                      (6,5,3,1)  (8,5,3,2)  (8,6,4,2)    (8,7,5,2)
                                 (8,6,3,1)  (8,7,4,1)    (12,7,2,1)
                                            (8,6,3,2,1)  (8,6,4,3,1)
                                                         (8,7,4,2,1)

This is the strict case of A045931, ranked by A350848 (zeros of A350941).
The conjugate version is A239241, non-strict A045931 (ranked by A325698).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, ranked by A349157, strict A352131.
- A277103, ranked by A350944.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 2    7        9        13        14         15         16
   --------------------------------------------------------------------
    (2)  (6,1)    (8,1)    (12,1)    (14)       (14,1)     (16)
         (4,2,1)  (4,3,2)  (6,4,3)   (6,5,3)    (6,5,4)    (8,5,3)
                  (6,2,1)  (8,3,2)   (10,3,1)   (8,4,3)    (12,3,1)
                           (10,2,1)  (6,4,3,1)  (10,3,2)   (6,5,4,1)
                                     (8,3,2,1)  (12,2,1)   (8,4,3,1)
                                                (6,5,3,1)  (10,3,2,1)
                                                           (6,4,3,2,1)

This is the strict case of A277579, ranked by A350943 (zeros of A350942).
The conjugate version is A352131, non-strict A277579 (ranked by A349157).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944, strict new.
- A350948, ranked by A350945, strict new.
There are three double-pairings of statistics:
- A351976, ranked by A350949, strict A010054?
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980. strict A014105?
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      10         14         18         21             24
   ----------------------------------------------------------------------
    (2,1)  (6,4)      (8,6)      (10,8)     (11,10)        (8,7,5,4)
           (4,3,2,1)  (5,4,3,2)  (6,5,4,3)  (8,6,4,3)      (9,8,4,3)
                      (6,5,2,1)  (7,6,3,2)  (8,7,4,2)      (10,8,4,2)
                                 (8,7,2,1)  (10,8,2,1)     (10,9,3,2)
                                            (6,5,4,3,2,1)  (11,10,2,1)
                                                           (8,6,4,3,2,1)

This is the strict case of A277579, ranked by A349157 (zeros of A350849).
The conjugate version is A352130, non-strict A277579 (ranked by A350943).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350942, A350950, A350951.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,30}]

A349150 Heinz numbers of integer partitions with at most one odd part.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 23, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 77, 78, 79, 81, 83, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109

Offset: 1

Gus Wiseman, Nov 10 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at most one odd prime index.

Also Heinz numbers of partitions with conjugate alternating sum <= 1.

			The terms and their prime indices begin:
      1: {}          23: {9}         49: {4,4}
      2: {1}         26: {1,6}       51: {2,7}
      3: {2}         27: {2,2,2}     53: {16}
      5: {3}         29: {10}        54: {1,2,2,2}
      6: {1,2}       31: {11}        57: {2,8}
      7: {4}         33: {2,5}       58: {1,10}
      9: {2,2}       35: {3,4}       59: {17}
     11: {5}         37: {12}        61: {18}
     13: {6}         38: {1,8}       63: {2,2,4}
     14: {1,4}       39: {2,6}       65: {3,6}
     15: {2,3}       41: {13}        67: {19}
     17: {7}         42: {1,2,4}     69: {2,9}
     18: {1,2,2}     43: {14}        71: {20}
     19: {8}         45: {2,2,3}     73: {21}
     21: {2,4}       47: {15}        74: {1,12}

The case of no odd parts is A066207, counted by A000041 up to 0's.
Requiring all odd parts gives A066208, counted by A000009.
These partitions are counted by A100824, even-length case A349149.
These are the positions of 0's and 1's in A257991.
The conjugate partitions are ranked by A349151.
The case of one odd part is A349158, counted by A000070 up to 0's.
A056239 adds up prime indices, row sums of A112798.
A122111 is a representation of partition conjugation.
A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A340932 ranks partitions whose least part is odd, counted by A026804.
A345958 ranks partitions with alternating sum 1.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[Reverse[primeMS[#]],_?OddQ]<=1&]

Union of A066207 (no odd parts) and A349158 (one odd part).

A349151 Heinz numbers of integer partitions with alternating sum <= 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 15, 16, 18, 24, 25, 32, 35, 36, 49, 50, 54, 60, 64, 72, 77, 81, 96, 98, 100, 121, 128, 135, 140, 143, 144, 150, 162, 169, 196, 200, 216, 221, 225, 240, 242, 256, 288, 289, 294, 308, 315, 323, 324, 338, 361, 375, 384, 392, 400, 437, 441, 450

Offset: 1

Gus Wiseman, Nov 10 2021

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 alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so these are also Heinz numbers of partitions with at most one odd conjugate part.

			The terms and their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   25: {3,3}
   32: {1,1,1,1,1}
   35: {3,4}
   36: {1,1,2,2}
   49: {4,4}

The case of alternating sum 0 is A000290.
These partitions are counted by A100824.
These are the positions of 0's and 1's in A344616.
The case of alternating sum 1 is A345958.
The conjugate partitions are ranked by A349150.
A000041 counts integer partitions.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A106529 ranks balanced partitions, counted by A047993.
A122111 is a representation of partition conjugation.
A257991 counts odd prime indices.
A316524 gives the alternating sum of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000070, A000700, A001222, A027187, A027193, A215366, A277103, A277579, A326841, A349149, A349158.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[Reverse[primeMS[#]]]<=1&]

Equals A000290 \/ A345958 decapitated.

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

cp's OEIS Frontend

A349158 Heinz numbers of integer partitions with exactly one odd part.

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A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

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A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

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A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

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A349150 Heinz numbers of integer partitions with at most one odd part.

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A349151 Heinz numbers of integer partitions with alternating sum <= 1.

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A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

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