A325406 -id:A325406 - cp's OEIS frontend

A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

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

First differs from A325388 in lacking 130.
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 A325404.

			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, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.

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

A325404 Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 5, 7, 5, 11, 12, 11, 12, 20, 15, 24, 22, 27, 28, 37, 28, 45, 43, 48, 50, 66, 58, 79, 72, 84, 87, 112, 106, 135, 128, 158, 147, 186, 180, 218, 220, 265, 246, 304, 303, 354, 340, 412, 418, 471, 463, 538, 543, 642, 600, 711, 755

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 A325405.

			The a(1) = 1 through a(12) = 5 reversed partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)   (B)    (C)
                 (13)  (14)  (15)  (16)  (17)  (18)  (19)  (29)   (39)
                       (23)        (25)  (26)  (27)  (28)  (38)   (57)
                                   (34)  (35)  (45)  (37)  (47)   (1B)
                                                     (46)  (56)   (2A)
                                                           (1A)
                                                           (146)

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

Cf. A279945, A325325, A325349, A325353, A325354, A325365, A325368, A325391, A325393, A325405, A325406, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],UnsameQ@@Join@@Table[Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 19, 24, 31, 26, 40, 43, 51, 52, 72, 66, 89, 88, 111, 119, 150, 130, 183, 193, 229, 231, 279, 287, 358, 365, 430, 426, 538, 535, 649, 680, 742, 803, 943, 982, 1136, 1115

Offset: 0

Gus Wiseman, May 03 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 A325467.

			The a(1) = 1 through a(9) = 6 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)
                                    (421)  (431)  (81)
                                           (521)  (621)

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

Cf. A000009, A325324, A325325, A325349, A325353, A325354, A325391, A325393, A325404, A325406, A325467.

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

A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 19, 24, 25, 36, 37, 43, 54, 63, 64, 80, 81, 100, 113, 122, 123, 151, 166, 178, 195, 217, 218, 269, 270, 295, 316, 332, 372, 424, 425, 447, 472, 547, 550, 616, 617, 659, 750, 777, 782, 862, 885, 995, 1032, 1083, 1090, 1176, 1275

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 A325400.

			The a(1) = 1 through a(8) = 15 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (112)   (113)    (33)      (34)       (35)
                    (1111)  (1112)   (114)     (115)      (44)
                            (11111)  (123)     (124)      (116)
                                     (222)     (223)      (125)
                                     (1113)    (1114)     (224)
                                     (11112)   (11113)    (1115)
                                     (111111)  (111112)   (1124)
                                               (1111111)  (2222)
                                                          (11114)
                                                          (111113)
                                                          (1111112)
                                                          (11111111)

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

Cf. A007294, A240026, A325353, A325356, A325360, A325362, A325391, A325393, A325394, A325400, A325404, A325406, A325468.

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

A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 13, 15, 19, 20, 20, 28, 28, 30, 36, 40, 40, 50, 50, 56, 64, 68, 68, 86, 86, 92, 102, 112, 114, 133, 133, 146, 158, 173, 173, 202, 202, 215, 237, 256, 256, 287, 287, 324, 340, 359, 359, 403, 423, 446, 464, 495, 495

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 A325398.

			The a(1) = 1 through a(9) = 6 reversed partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (12)  (13)  (14)  (15)  (16)   (17)   (18)
                        (23)  (24)  (25)   (26)   (27)
                                    (34)   (35)   (36)
                                    (124)  (125)  (45)
                                                  (126)
The smallest reversed strict partition with strictly increasing differences not counted by this sequence is (1,2,4,7), whose first and second differences are (1,2,3) and (1,1) respectively.

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

Cf. A179254, A240026, A325353, A325354, A325357, A325393, A325395, A325398, A325404, A325406, A325456, A325468.

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

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}]

A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing 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, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A301899 in lacking 105. First differs from A325399 in having 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 A325391.

			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, A325357, A325391, A325395, A325397, A325399, A325400, A325405, A325406, A325456, A325467.

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

A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 11, 12, 13, 17, 16, 19, 23, 23, 24, 30, 29, 35, 37, 37, 40, 49, 47, 51, 56, 59, 61, 73, 65, 75, 80, 84, 91, 99, 91, 103, 112, 120, 114, 132, 126, 143, 154, 147, 152, 175, 169, 190, 187, 194, 198, 226, 225, 231, 236, 246, 256, 293

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 A325397.

			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)     (2221)     (332)
                                     (111111)  (1111111)  (431)
                                                          (2222)
                                                          (11111111)
The first partition that has weakly decreasing differences (A320466) but is not counted under a(9) is (3,3,2,1), 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. A320466, A320509, A325350, A325354, A325391, A325393, A325397, A325398, A325399, A325404, A325405, A325406, A325468.

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

A325466 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 5, 4, 0, 1, 0, 0, 1, 4, 6, 3, 0, 1, 0, 0, 1, 6, 6, 4, 3, 1, 1, 0, 0, 1, 6, 10, 4, 2, 4, 1, 2, 0, 0, 1, 7, 12, 8, 3, 3, 4, 1, 2, 1, 0, 1, 6, 13, 11, 2, 11, 3, 4, 0, 3, 1, 1, 1, 10, 16, 7, 10, 10

Offset: 0

Gus Wiseman, May 04 2019

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.

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  3  1  0  0
  1  3  2  1  0  0
  1  5  4  0  1  0  0
  1  4  6  3  0  1  0  0
  1  6  6  4  3  1  1  0  0
  1  6 10  4  2  4  1  2  0  0
  1  7 12  8  3  3  4  1  2  1  0
  1  6 13 11  2 11  3  4  0  3  1  1
  1 10 16  7 10 10  6  6  5  1  1  2  1
  1  7 18 14  7 16 11  6  4  8  0  5  0  1
  1  9 20 18 10 20 13 10 10  4  5  5  2  2  2
  1 10 26 18 10 24 13 19 13 10  6  6  2  8  1  2
  1 11 25 24 16 28 19 24 14 15  9 10  9  5  2  7  1
Row 7 counts the following reversed partitions (empty columns not shown):
  (7)  (16)       (115)     (133)   (11122)
       (25)       (124)     (1123)
       (34)       (223)     (1222)
       (1111111)  (1114)
                  (11113)
                  (111112)
Row 9 counts the following reversed partitions (empty columns not shown):
(9)  (18)         (117)       (126)    (1125)   (1134)    (11223)  (111222)
     (27)         (135)       (144)    (11124)  (1224)             (1111122)
     (36)         (225)       (1233)            (11133)
     (45)         (234)       (12222)           (111123)
     (333)        (1116)
     (111111111)  (2223)
                  (11115)
                  (111114)
                  (1111113)
                  (11111112)

Row sums are A000041. Column k = 1 is A088922.

Cf. A098859, A279945, A325242, A325324, A325325, A325349, A325404, A325405, A325406, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,1,Length[#]}]]==k&]],{n,0,16},{k,0,n}]

A355523 Number of distinct differences between adjacent prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 0, 2, 0, 1, 2, 2, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 1, 2, 0, 2, 2, 2, 0, 2, 0, 2, 1

Offset: 1

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			For example, the prime indices of 22770 are {1,2,2,3,5,9}, with differences (1,0,1,2,4), so a(22770) = 4.

Crossrefs found in the link are not repeated here.
Counting m such that A056239(m) = n and a(m) = k gives A279945.
With multiplicity we have A252736(n) = A001222(n) - 1.
The maximal difference is A286470, minimal A355524.
A008578 gives the positions of 0's.
A287352 lists differences between 0-prepended prime indices.
A355534 lists augmented differences between prime indices.
A355536 lists differences between prime indices.
Cf. A066312, A238353, A238710, A286469, A320348, A325388, A325406, A351294, A352827, A355525-A355533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Differences[primeMS[n]]]],{n,1000}]

A355523(n) = if(1==n, 0, my(pis = apply(primepi,factor(n)[,1]), difs = vector(#pis-1, i, pis[i+1]-pis[i])); (#Set(difs)+!issquarefree(n))); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

cp's OEIS Frontend

A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

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A325404 Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

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A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

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A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

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A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

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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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A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

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A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

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