A325368 -id:A325368 - cp's OEIS frontend

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

325, 845, 931, 1625, 2527, 3509, 6253, 6517, 8125, 9251

Offset: 1

Gus Wiseman, May 18 2025

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.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
A integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its positive 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
   325: {3,3,6}
   845: {3,6,6}
   931: {4,4,8}
  1625: {3,3,3,6}
  2527: {4,8,8}
  3509: {5,5,10}
  6253: {6,6,12}
  6517: {4,4,4,8}
  8125: {3,3,3,3,6}
  9251: {5,10,10}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383511.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A047966, A051903, A212166, A238745, A325368, A351592, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]

A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 15 2025

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.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Partitions of this type are counted by A383507.
Negating both sides gives A383531, counted by A383530.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
paug[y_]:=-DeleteCases[Differences[Append[y,0]],0];
Select[Range[100], UnsameQ@@Last/@FactorInteger[#] && UnsameQ@@paug[Reverse[prix[#]]]&]

Equals A130091 /\ A383512.

A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 2, 5, 1, 1, 6, 1, 3, 2, 1, 1, 7, 1, 1, 8, 1, 2, 2, 2, 1, 4, 9, 1, 1, 3, 1, 5, 2, 1, 3, 10, 1, 1, 1, 11, 1, 2, 3, 1, 6, 3, 1, 1, 1, 12, 1, 7, 2, 4, 1, 2, 13, 1, 1, 2, 14, 1, 4, 2, 1, 1, 8, 15, 1, 1, 4, 1, 2, 2, 5, 1, 5, 16, 1, 1, 3, 2

Offset: 1

Gus Wiseman, May 20 2025

Also differences of distinct 0-prepended prime indices of n.

			The prime indices of 140 are {1,1,3,4}, zero prepended {0,1,1,3,4}, differences (1,0,2,1), positive (1,2,1).
Rows begin:
    1: ()        16: (1)        31: (11)
    2: (1)       17: (7)        32: (1)
    3: (2)       18: (1,1)      33: (2,3)
    4: (1)       19: (8)        34: (1,6)
    5: (3)       20: (1,2)      35: (3,1)
    6: (1,1)     21: (2,2)      36: (1,1)
    7: (4)       22: (1,4)      37: (12)
    8: (1)       23: (9)        38: (1,7)
    9: (2)       24: (1,1)      39: (2,4)
   10: (1,2)     25: (3)        40: (1,2)
   11: (5)       26: (1,5)      41: (13)
   12: (1,1)     27: (2)        42: (1,1,2)
   13: (6)       28: (1,3)      43: (14)
   14: (1,3)     29: (10)       44: (1,4)
   15: (2,1)     30: (1,1,1)    45: (2,1)

Row-lengths are A001221, sums A061395.
Rows start with A055396, end with A241919.
For multiplicities instead of differences we have A124010 (prime signature).
Including difference 0 gives A287352, without prepending A355536.
Positions of first appearances of rows are A358137.
Positions of strict rows are A383512, counted by A098859.
Positions of non-strict rows are A383513, counted by A336866.
Heinz numbers of rows are A383535.
Restricting to rows of squarefree index gives A384008.
Without prepending we get A384009.
A000040 lists the primes, differences A001223.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A005117, A122111, A130091, A325325, A325349, A325366, A325368, A381431, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]

a(A005117(n)) = A384008(n).

A325467 Heinz numbers of integer partitions y such that the k-th differences of y are distinct (independently) 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, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107

Offset: 1

Gus Wiseman, May 03 2019

nonn

First differs from A301899 in having 70 and lacking 105.
First differs from A325398 in having 70.
First differs from A319315 in having 966.
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 A325468.

			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}
For example, the k-th differences for k = 0...3 of the partition (9,4,2,1) with Heinz number 966 are
   9  4  2  1
  -5 -2 -1
   3  1
  -2
and since the entries of each row are distinct, 966 belongs to the sequence.

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

A subsequence of A005117.

Cf. A056239, A112798, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325405, A325468.

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

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713

Offset: 0

Gus Wiseman, May 14 2025

nonn

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

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

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#] && UnsameQ@@DeleteCases[Differences[Append[#,0]],0]&]],{n,0,30}]

These partitions have Heinz numbers A130091 /\ A383512.

A383514 Heinz numbers of non Wilf section-sum partitions.

Original entry on oeis.org

10, 14, 15, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190

Offset: 1

Gus Wiseman, May 18 2025

nonn

First differs from A384007 in having 1000.
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.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The terms together with their prime indices begin:
    10: {1,3}    57: {2,8}      94: {1,15}
    14: {1,4}    58: {1,10}     95: {3,8}
    15: {2,3}    62: {1,11}    100: {1,1,3,3}
    22: {1,5}    65: {3,6}     106: {1,16}
    26: {1,6}    69: {2,9}     111: {2,12}
    33: {2,5}    74: {1,12}    115: {3,9}
    34: {1,7}    77: {4,5}     118: {1,17}
    35: {3,4}    82: {1,13}    119: {4,7}
    38: {1,8}    85: {3,7}     122: {1,18}
    39: {2,6}    86: {1,14}    123: {2,13}
    46: {1,9}    87: {2,10}    129: {2,14}
    51: {2,7}    91: {4,6}     130: {1,3,6}
    55: {3,5}    93: {2,11}    133: {4,8}

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A351592 (A384006).
These partitions are counted by A383506.
The Look-and-Say case is A383511 (A383518).
For Wilf instead of non Wilf we have A383519 (A383520).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

A325552 Number of compositions of n with distinct differences up to sign.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 12, 23, 38, 61, 78, 135, 194, 315, 454, 699, 982, 1495, 2102, 3085, 4406, 6583, 9048, 13117, 18540, 26399, 36484, 51885, 72498, 100031, 139342, 192621, 267068, 367631, 505954, 687153, 946412, 1283367, 1745974, 2356935, 3207554, 4311591, 5816404

Offset: 0

Gus Wiseman, May 11 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).
a(n) has the same parity as n for n > 0, since reversing a composition does not change whether or not it has this property, and the only valid symmetric compositions are (n) and (n/2,n/2), with the latter only existing for even n. - Charlie Neder, Jun 06 2019

			The differences of (1,2,1) are (1,-1), which are different but not up to sign, so (1,2,1) is not counted under a(4).
The a(1) = 1 through a(7) = 23 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)
       (11)  (12)  (13)   (14)   (15)   (16)
             (21)  (22)   (23)   (24)   (25)
                   (31)   (32)   (33)   (34)
                   (112)  (41)   (42)   (43)
                   (211)  (113)  (51)   (52)
                          (122)  (114)  (61)
                          (221)  (132)  (115)
                          (311)  (213)  (124)
                                 (231)  (133)
                                 (312)  (142)
                                 (411)  (214)
                                        (223)
                                        (241)
                                        (322)
                                        (331)
                                        (412)
                                        (421)
                                        (511)
                                        (1132)
                                        (2113)
                                        (2311)
                                        (3112)

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

Cf. A011782, A070211, A175342, A242882, A325325, A325368, A325404, A325545, A325551, A325553, A325555, A325557.

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

a(26)-a(42) from Alois P. Heinz, Jan 27 2024

A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 20, 23, 25, 28, 29, 31, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 121, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164

Offset: 1

Gus Wiseman, May 15 2025

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.

Integer partitions with distinct multiplicities are called Wilf partitions.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}
   47: {15}
   49: {4,4}
   50: {1,3,3}

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

For just distinct multiplicities we have A130091 (conjugate A383512), counted by A098859.
For just distinct 0-appended differences we have A325367, counted by A325324.
These partitions are counted by A383709.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
Cf. A000720, A005117, A047966, A238745, A320348, A325325, A325349, A325355, A325366, A325368, A325388, A383506.

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

Equals A130091 /\ A325367.

A342521 Heinz numbers of integer partitions with distinct first quotients.

Original entry on oeis.org

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, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82

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 1365 are {2,3,4,6}, with first quotients (3/2,4/3,3/2), so 1365 is not 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}
   16: {1,1,1,1}
   24: {1,1,1,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
  100: {1,1,3,3}

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

For multiplicities (prime signature) instead of quotients we have A130091.
For differences instead of quotients we have A325368 (count: A325325).
These partitions are counted by A342514 (strict: A342520, ordered: A342529).
The equal instead of distinct version is A342522.
The version counting strict divisor chains is A342530.
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.
Cf. A003242, A005117, A056239, A067824, A098859, A112798, A169594, A253249, A325326, A325337, A325405.

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

A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 6, 11, 4, 13, 10, 6, 2, 17, 4, 19, 6, 9, 14, 23, 4, 5, 22, 3, 10, 29, 8, 31, 2, 15, 26, 10, 4, 37, 34, 21, 6, 41, 12, 43, 14, 6, 38, 47, 4, 7, 6, 33, 22, 53, 4, 15, 10, 39, 46, 59, 8, 61, 58, 9, 2, 25, 20, 67, 26, 51, 12, 71, 4, 73

Offset: 1

Gus Wiseman, May 21 2025

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.

Also Heinz number of the first differences of the distinct 0-prepended prime indices of n.

			The terms together with their prime indices begin:
     1: {}        2: {1}        31: {11}       38: {1,8}
     2: {1}      17: {7}         2: {1}        47: {15}
     3: {2}       4: {1,1}      15: {2,3}       4: {1,1}
     2: {1}      19: {8}        26: {1,6}       7: {4}
     5: {3}       6: {1,2}      10: {1,3}       6: {1,2}
     4: {1,1}     9: {2,2}       4: {1,1}      33: {2,5}
     7: {4}      14: {1,4}      37: {12}       22: {1,5}
     2: {1}      23: {9}        34: {1,7}      53: {16}
     3: {2}       4: {1,1}      21: {2,4}       4: {1,1}
     6: {1,2}     5: {3}         6: {1,2}      15: {2,3}
    11: {5}      22: {1,5}      41: {13}       10: {1,3}
     4: {1,1}     3: {2}        12: {1,1,2}    39: {2,6}
    13: {6}      10: {1,3}      43: {14}       46: {1,9}
    10: {1,3}    29: {10}       14: {1,4}      59: {17}
     6: {1,2}     8: {1,1,1}     6: {1,2}       8: {1,1,1}

For multiplicities instead of differences we have A181819.
Positions of first appearances are A358137.
Positions of squarefree numbers are A383512, counted by A098859.
Positions of nonsquarefree numbers are A383513, counted by A336866.
These are Heinz numbers of rows of A383534.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A122111, A124010 (prime signature), A130091, A287352, A325351, A325366, A325368, A355536, A381431, A384008, A384009.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]

A001222(a(n)) = A001221(n).
A056239(a(n)) = A061395(n).
A055396(a(n)) = A055396(n).
A061395(a(n)) = A241919(n).

cp's OEIS Frontend

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

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A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

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A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

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A325467 Heinz numbers of integer partitions y such that the k-th differences of y are distinct (independently) for all k >= 0.

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A383507 Number of Wilf and conjugate Wilf integer partitions of n.

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A383514 Heinz numbers of non Wilf section-sum partitions.

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A325552 Number of compositions of n with distinct differences up to sign.

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A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

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