A325364 -id:A325364 - cp's OEIS frontend

A325367 Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325324.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  28: {1,1,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Positions of squarefree numbers in A325390.

Cf. A056239, A112798, A130091, A320348, A325324, A325327, A325362, A325364, A325366, A325368, A325388, A325405, A325407, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[200],UnsameQ@@Differences[Append[primeptn[#],0]]&]

A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A007294.
This sequence and A025487, considered as sets, are related by the partition conjugation function A122111(.), which maps the members of either set 1:1 onto the other set. - Peter Munn, Feb 10 2022

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   30: {1,2,3}
   31: {11}
   33: {2,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A056239, A112798, A240026, A320348, A325327, A325360, A325364, A325367, A325390, A325394, A325400.

Related to A025487 via A122111.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],OrderedQ[Differences[Append[primeptn[#],0]]]&]

A325388 Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A320348.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   35: {3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A056239, A112798, A320348, A325324, A325327, A325362, A325364, A325366, A325367, A325368, A325390, A325405, A325460, A325461, A325467.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],SquareFreeQ[#]&&UnsameQ@@Differences[Append[primeptn[#],0]]&]

A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 10, 15, 13, 18, 17, 21, 15, 24, 19, 18, 23, 30, 25, 33, 29, 36, 14, 39, 20, 42, 31, 27, 37, 48, 35, 51, 21, 36, 41, 57, 55, 60, 43, 45, 47, 66, 30, 69, 53, 72, 22, 30, 65, 78, 59, 36, 35, 84, 85, 87, 61, 54, 67, 93, 50, 96, 49, 63, 71

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

			The Heinz number of (6,3,1) is 130, and its negated differences plus one are (4,3,2), which has Heinz number 105, so a(130) = 105.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Number of appearances of n is A325392(n).
Positions of squarefree numbers are A325367.
Cf. A007294, A007862, A056239, A112798, A320509, A325324, A325327, A325351, A325352, A325362, A325364, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Times@@Prime/@(1-Differences[Append[primeptn[n],0]]),{n,100}]

A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169

Offset: 0

Seiichi Manyama, Oct 14 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences (with the first part taken to be 0) of (6,3,1) are (-3,-2,-1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are weakly decreasing. The Heinz numbers of these partitions are given by A325364. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences (with the first part taken to be 0) are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(11) = 8 such partitions of 11:
01: [11]
02: [4, 7]
03: [5, 6]
04: [2, 4, 5]
05: [3, 4, 4]
06: [2, 3, 3, 3]
07: [1, 2, 2, 2, 2, 2]
08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(12) = 11 such partitions of 12:
01: [12]
02: [4, 8]
03: [5, 7]
04: [6, 6]
05: [2, 4, 6]
06: [3, 4, 5]
07: [4, 4, 4]
08: [3, 3, 3, 3]
09: [1, 2, 3, 3, 3]
10: [2, 2, 2, 2, 2, 2]
11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..300 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026, A240027, A320466, A320470, A320510.
Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).
Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0
  }
  cnt
end
def A320509(n)
  (0..n).map{|i| f(i)}
end
p A320509(50)

A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 80, 82, 83

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325350.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A093641, A112798, A320466, A320509, A325350, A325351, A325361, A325364, A325366, A325394, A325395, A325396, A325397.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A325361 Heinz numbers of integer partitions whose differences are weakly decreasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320466.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],GreaterEqual@@Differences[primeptn[#]]&]

A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325389.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (52)       (53)
             (111)  (211)   (41)     (51)      (61)       (62)
                    (1111)  (311)    (321)     (421)      (71)
                            (2111)   (411)     (511)      (521)
                            (11111)  (3111)    (3211)     (611)
                                     (21111)   (4111)     (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.

Mathematica
```
aug[y_]:=Table[If[i
				
```

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

A325460 Heinz numbers of integer partitions with strictly increasing differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 122, 123, 127, 129, 130, 131

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A179269.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   37: {12}
   38: {1,8}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A007294, A056239, A112798, A179269, A325327, A325362, A325364, A325367, A325388, A325390, A325395, A325398, A325456, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Differences[Append[primeptn[#],0]]&]

A325397 Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A325361 in lacking 150.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325353.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
The first partition that has weakly decreasing differences (A320466, A325361) but is not represented in this sequence is (3,3,2,1), which has Heinz number 150 and whose first and second differences are (0,-1,-1) and (-1,0) respectively.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[GreaterEqual@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

cp's OEIS Frontend

A325367 Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).

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A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

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A325388 Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).

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A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

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A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

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A325389 Heinz numbers of integer partitions whose augmented differences are weakly decreasing.

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A325361 Heinz numbers of integer partitions whose differences are weakly decreasing.

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A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

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