A320466 -id:A320466 - cp's OEIS frontend

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

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[#]}]&]

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[#]}]&]

A320510 Number of partitions of n such that the successive differences of consecutive parts are decreasing, and first difference < first part.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 3, 4, 3, 4, 6, 3, 5, 6, 5, 6, 9, 5, 7, 9, 8, 8, 11, 8, 11, 13, 10, 12, 15, 11, 15, 16, 14, 16, 21, 15, 20, 22, 18, 21, 26, 21, 24, 28, 25, 28, 33, 26, 32, 34, 33, 36, 40, 34, 40, 45, 40, 43, 49, 43, 52, 54, 49, 54, 62, 56, 62, 64, 61, 67, 75, 66

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 of (6,3,1) (with the last part taken to be 0) 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 strictly decreasing. The Heinz numbers of these partitions are given by A325461. 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 04 2019

			There are a(29) = 13 such partitions of 29:
01: [29]
02: [10, 19]
03: [11, 18]
04: [12, 17]
05: [13, 16]
06: [14, 15]
07: [6, 10, 13]
08: [6, 11, 12]
09: [7, 10, 12]
10: [7, 11, 11]
11: [8, 10, 11]
12: [9, 10, 10]
13: [4, 7, 9, 9]
There are a(30) = 10 such partitions of 30:
01: [30]
02: [11, 19]
03: [12, 18]
04: [13, 17]
05: [14, 16]
06: [15, 15]
07: [6, 11, 13]
08: [7, 11, 12]
09: [8, 11, 11]
10: [4, 7, 9, 10]

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, A320509.
Cf. A320385 (distinct parts, decreasing, and first difference < first part).
Cf. A007294, A007862, A325324, A325358, A325393, A325461.

Table[Length[Select[IntegerPartitions[n],Greater@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 04 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 && ary0.uniq == ary0
  }
  cnt
end
def A320510(n)
  (0..n).map{|i| f(i)}
end
p A320510(50)

A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 2, 4, 3, 1, 7, 1, 3, 8, 4, 1, 10, 1, 6, 10, 4, 1, 14, 4, 4, 12, 7, 1, 19, 1, 6, 14, 5, 7, 22, 1, 5, 16, 12, 1, 24, 1, 8, 25, 6, 1, 27, 4, 12, 21, 9, 1, 29, 9, 12, 23, 7, 1, 40, 1, 7, 30, 11, 10, 35, 1, 10, 27, 21, 1, 42, 1, 8, 39, 11, 7, 40, 1, 22, 35, 9, 1, 49, 12, 9, 34

Offset: 1

Clark Kimberling

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12580
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Jon Maiga, Computer-generated formulas for A049992, Sequence Machine.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A007862, A014405, A023645, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049994, A175676, A240026, A240027, A307824, A320466.

A049992(n) = (A014405(n)+A023645(n)); \\ (Uses the programs given in A014405 and A023645) - Antti Karttunen, Feb 20 2023

G.f.: Sum_{k>=3} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988] - Petros Hadjicostas, Sep 29 2019

a(n) = A014405(n) + A023645(n) = A049994(n) + A175676(n). [Two of the formulas listed by Sequence Machine, both obviously true] - Antti Karttunen, Feb 20 2023

More terms from Petros Hadjicostas, Sep 29 2019

A342513 Number of integer partitions of n with weakly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 15, 20, 21, 24, 28, 29, 33, 40, 44, 49, 57, 61, 65, 77, 84, 87, 99, 106, 115, 132, 141, 152, 167, 180, 193, 212, 228, 246, 274, 290, 309, 338, 357, 382, 412, 439, 463, 498, 536, 569, 608, 648, 693, 743, 790, 839, 903, 949

Offset: 1

Gus Wiseman, Mar 17 2021

nonn

Also called log-concave-down partitions.
Also the number of reversed integer partitions of n with weakly decreasing first quotients.
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 partition (9,7,4,2,1) has first quotients (7/9,4/7,1/2,1/2) so is counted under a(23).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (111111)  (2221)     (431)
                                               (1111111)  (2222)
                                                          (11111111)

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.

The ordered version is A069916.
The version for differences instead of quotients is A320466.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The strict case is A342519.
The Heinz numbers of these partitions are A342526.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with adjacent parts x <= 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

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, 42, 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

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-down partitions.
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 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
Most small numbers are in the sequence, but 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}
   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}
   84: {1,1,2,4}

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.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A242031.
For differences instead of quotients we have A325361 (count: A320466).
These partitions are counted by A342513 (strict: A342519, ordered: A069916).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A002843 counts compositions with all adjacent parts x <= 2y.
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.
Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.

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

A049993 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 7, 9, 13, 16, 17, 24, 25, 28, 36, 40, 41, 51, 52, 58, 68, 72, 73, 87, 91, 95, 107, 114, 115, 134, 135, 141, 155, 160, 167, 189, 190, 195, 211, 223, 224, 248, 249, 257, 282, 288, 289, 316, 320, 332, 353, 362, 363, 392, 401, 413, 436, 443, 444, 484, 485, 492, 522, 533, 543, 578

Offset: 1

Clark Kimberling

nonn

Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A007862, A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A240026, A240027, A307824, A320466, A325325.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049992(k).
G.f.: (g.f. of A049992)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

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

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A320510 Number of partitions of n such that the successive differences of consecutive parts are decreasing, and first difference < first part.

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A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

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A342513 Number of integer partitions of n with weakly decreasing first quotients.

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A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

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A049993 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum <= n.

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