A325457 -id:A325457 - cp's OEIS frontend

A325368 Heinz numbers of integer partitions with distinct differences between successive parts.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 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 enumeration of these partitions by sum is given by A325325.

			Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   88: {1,1,1,5}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  100: {1,1,3,3}

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

Cf. A056239, A112798, A130091, A240026, A325325, A325328, A325352, A325360, A325361, A325366, A325367, A325405, A325456, A325457.

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

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[#]]&]

A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 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, 111, 113, 114, 115

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 A325358.

			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}
   31: {11}
   33: {2,5}
   34: {1,7}

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

A subsequence of A005117.

Cf. A056239, A093641, A112798, A320466, A325351, A325358, A325366, A325389, A325393, A325394, A325395, A325457.

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

A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 7, 6, 8, 10, 10, 11, 14, 13, 16, 19, 18, 20, 25, 23, 27, 31, 30, 34, 39, 37, 42, 48, 47, 50, 59, 56, 63, 70, 68, 74, 83, 82, 89, 97, 97, 104, 116, 113, 123, 133, 133, 142, 155, 153, 166, 178, 178, 189, 204, 204, 218, 232, 235, 247, 265, 265, 283, 299

Offset: 0

Seiichi Manyama, Oct 13 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) > p(k) - p(k-1) for all k >= 3.
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). Then a(n) is the number of integer partitions of n whose differences are strictly decreasing. The Heinz numbers of these partitions are given by A325457. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(10) = 8 such partitions of 10:
01: [10]
02: [1, 9]
03: [2, 8]
04: [3, 7]
05: [4, 6]
06: [5, 5]
07: [1, 4, 5]
08: [2, 4, 4]
There are a(11) = 10 such partitions of 11:
01: [11]
02: [1, 10]
03: [2, 9]
04: [3, 8]
05: [4, 7]
06: [5, 6]
07: [1, 4, 6]
08: [1, 5, 5]
09: [2, 4, 5]
10: [3, 4, 4]

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

Cf. A049988, A240026, A240027, A320466, A320510, A325325, A325358, A325393, A325457.

Table[Length[Select[IntegerPartitions[n],Greater@@Differences[#]&]],{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|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
  }
  cnt
end
def A320470(n)
  (0..n).map{|i| f(i)}
end
p A320470(50)

A325548 Number of compositions of n with strictly decreasing differences.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 10, 13, 19, 23, 29, 38, 46, 55, 69, 80, 96, 115, 132, 154, 183, 207, 238, 276, 314, 356, 405, 455, 513, 579, 647, 724, 809, 897, 998, 1107, 1225, 1350, 1486, 1639, 1805, 1973, 2166, 2374, 2586, 2824, 3084, 3346, 3646, 3964, 4286, 4655, 5047

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)
       (11)  (12)  (13)   (14)   (15)    (16)   (17)
             (21)  (22)   (23)   (24)    (25)   (26)
                   (31)   (32)   (33)    (34)   (35)
                   (121)  (41)   (42)    (43)   (44)
                          (122)  (51)    (52)   (53)
                          (131)  (132)   (61)   (62)
                          (221)  (141)   (133)  (71)
                                 (231)   (142)  (134)
                                 (1221)  (151)  (143)
                                         (232)  (152)
                                         (241)  (161)
                                         (331)  (233)
                                                (242)
                                                (251)
                                                (332)
                                                (341)
                                                (431)
                                                (1331)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A011782, A000740, A008965, A070211, A175342, A179254, A320470, A325457, A325545, A325546, A325547, A325552, A325557.

b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or
       j-l b(n, 0$2):
seq(a(n), n=0..52);  # Alois P. Heinz, Jan 27 2024

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,15}]

a(26)-a(44) from Lars Blomberg, May 30 2019

a(45)-a(52) from Alois P. Heinz, Jan 27 2024

A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 13, 16, 19, 18, 20, 24, 22, 26, 29, 28, 31, 37, 33, 38, 43, 42, 44, 52, 48, 55, 59, 58, 62, 72, 65, 74, 80, 80, 82, 94, 88, 99, 103, 104, 108, 123, 114, 126, 133, 135, 137, 155, 145, 161, 166, 169, 174

Offset: 0

Gus Wiseman, May 02 2019

nonn

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 Heinz numbers of these partitions are given by A325399.

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)    (9)
            (21)  (31)  (32)  (42)  (43)  (53)   (54)
                        (41)  (51)  (52)  (62)   (63)
                                    (61)  (71)   (72)
                                          (431)  (81)

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

Cf. A049988, A320466, A325353, A325354, A325358, A325391, A325396, A325399, A325404, A325406, A325457, A325468.

Table[Length[Select[IntegerPartitions[n],And@@Table[Greater@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325399 Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A167171 in having 70. First differs from A325398 in lacking 42.
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 A325393.

			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}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}

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

A subsequence of A005117.

Cf. A056239, A112798, A320466, A320470, A325358, A325391, A325393, A325396, A325397, A325398, A325400, A325405, A325457, A325467.

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

A325456 Heinz numbers of integer partitions with strictly increasing differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87

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) are (-3,-2).
The enumeration of these partitions by sum is given by A240027.

			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}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

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

Cf. A056239, A112798, A179269, A240026, A240027, A325328, A325352, A325360, A325361, A325368, A325395, A325398, A325457, A325460.

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

A325461 Heinz numbers of integer partitions with strictly decreasing differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 75, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197

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 A320510.

			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}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

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

Cf. A056239, A112798, A320510, A325327, A325362, A325364, A325367, A325388, A325390, A325396, A325399, A325407, A325457, A325460.

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

A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Mar 23 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 first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 150 are {1,2,3,3}, with first quotients (2,3/2,1), so 150 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   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}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

For multiplicities (prime signature) instead of quotients we have A304686.
For differences instead of quotients we have A325457 (count: A320470).
The version counting strict divisor chains is A342086.
These partitions are counted by A342499 (strict: A342518, ordered: A342494).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A056239, A067824, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A325405, A334997, A342530.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

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A325368 Heinz numbers of integer partitions with distinct differences between successive parts.

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

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A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.

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A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing.

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A325548 Number of compositions of n with strictly decreasing differences.

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A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

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A325399 Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.

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A325456 Heinz numbers of integer partitions with strictly increasing differences.

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