A325352 -id:A325352 - cp's OEIS frontend

A359360 Length times minimum part of the integer partition with Heinz number n. Least prime index of n times number of prime indices of n.

0, 1, 2, 2, 3, 2, 4, 3, 4, 2, 5, 3, 6, 2, 4, 4, 7, 3, 8, 3, 4, 2, 9, 4, 6, 2, 6, 3, 10, 3, 11, 5, 4, 2, 6, 4, 12, 2, 4, 4, 13, 3, 14, 3, 6, 2, 15, 5, 8, 3, 4, 3, 16, 4, 6, 4, 4, 2, 17, 4, 18, 2, 6, 6, 6, 3, 19, 3, 4, 3, 20, 5, 21, 2, 6, 3, 8, 3, 22, 5, 8, 2

Offset: 1

Gus Wiseman, Dec 28 2022

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n.

			The partition with Heinz number 7865 is (6,5,5,3), so a(7865) = 4*3 = 12.

Difference of A056239 and A359358.
The opposite version is A326846.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, length A001222, sum A056239.
A243055 subtracts the least prime index from the greatest.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Cf. A006141, A124010, A246277, A268192, A316413, A325352, A326836, A326837, A326844, A355534, A356958.

Table[PrimeOmega[n]*PrimePi[FactorInteger[n][[1,1]]],{n,100}]

a(n) = if (n==1, 0, my(f=factor(n)); bigomega(f)*primepi(f[1, 1])); \\ Michel Marcus, Dec 28 2022

a(n) = A001222(n) * A055396(n).

A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-up 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 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}

Robert Price, Table of n, a(n) for n = 1..10000
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 A304678.
For differences instead of quotients we have A325360 (count: A240026).
These partitions are counted by A342523 (strict: A342516, ordered: A342492).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with adjacent parts x >= 2y.
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.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A000005, A002843, A056239, A067824, A112798, A124010, A130091, A238710, A253249, A325351, A325352, A342191.

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

A356958 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (b-a+1, ..., y-a+1, z-a+1).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 2, 4, 2, 1, 1, 1, 2, 2, 1, 3, 3, 5, 1, 1, 2, 1, 6, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 7, 2, 1, 2, 2, 8, 5, 1, 1, 3, 2, 4, 1, 5, 1, 2, 9, 1, 1, 1, 2, 1, 3, 3, 6, 1, 6, 2, 2, 2, 3, 1, 1, 4, 7, 10, 1, 2, 3, 11, 1, 3, 1, 1, 1, 1, 1, 4, 2, 5

Offset: 1

Gus Wiseman, Dec 27 2022

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.

			Triangle begins:
   1:   .
   2:   .
   3:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  1 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
For example, the prime indices of 315 are (2,2,3,4), so row 315 is (2,3,4) - 2 + 1 = (1,2,3).

Row lengths are A001222(n) - 1.
Indices of empty rows are A008578.
Even bisection is A112798.
Heinz numbers of rows are A246277.
An opposite version is A358172, Heinz numbers A358195.
Row sums are A359358(n) + A001222(n) - 1.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
Cf. A055396, A124010, A241916, A253565, A325351, A325352, A326844, A355534, A355536, A358137.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,{},1-First[primeMS[n]]+Rest[primeMS[n]]],{n,100}]

A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 2, 4, 2, 1, 1, 1, 2, 1, 3, 3, 3, 5, 2, 2, 2, 1, 6, 1, 1, 4, 4, 3, 2, 1, 1, 1, 1, 4, 7, 2, 2, 2, 1, 8, 5, 3, 3, 3, 4, 3, 5, 5, 2, 2, 9, 2, 2, 2, 2, 1, 3, 1, 6, 6, 6, 2, 1, 1, 3, 4, 4, 4, 7, 10, 3, 3, 2, 11, 3, 3, 1, 1, 1, 1, 1, 4, 5, 4

Offset: 1

Gus Wiseman, Dec 20 2022

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.

			Triangle begins:
   1:   .
   2:   .
   3:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  2 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
  17:   .
  18:  2 1
  19:   .
  20:  3 3
For example, the prime indices of 900 are (1,1,2,2,3,3), so row 900 is 3 - (1,1,2,2,3) + 1 = (3,3,2,2,1).

Row lengths are A001222(n) - 1.
Indices of empty rows are A008578.
Even-indexed rows have sums A243503.
Row sums are A326844(n) + A001222(n) - 1.
An opposite version is A356958, Heinz numbers A246277.
Heinz numbers of the rows are A358195, even bisection A241916.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
Cf. A055396, A124010, A253565, A325351, A325352, A355534, A355536, A358137.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,{},1+Last[primeMS[n]]-Most[primeMS[n]]],{n,100}]

A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

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, 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, 85, 86, 87, 89, 91

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 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 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}
   18: {1,2,2}
   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}
   42: {1,2,4}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

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 differences instead of quotients we have A325456 (count: A240027).
For multiplicities (prime signature) instead of quotients we have A334965.
The version counting strict divisor chains is A342086.
These partitions are counted by A342498 (strict: A342517, ordered: A342493).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
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. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

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

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]&]

A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 2, 0, 4, 0, 4, 1, 8, 0, 1, 0, 4, 5, 5, 0, 3, 2, 3, 6, 9, 0, 3, 0, 10, 2, 0, 3, 5, 0, 6, 7, 5, 0, 2, 0, 11, 2, 7, 1, 6, 0, 2, 0, 12, 0, 4, 4, 13

Offset: 1

Gus Wiseman, Dec 27 2022

nonn

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

			The partition with Heinz number 7865 is (6,5,5,3), which has the following diagram. The 3 X 4 rectangle is shown in dots.
  . . . o o o
  . . . o o
  . . . o o
  . . .
The size of the complement is 7, so a(7865) = 7.

The opposite version is A326844.
Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326846 = size of the smallest rectangle containing the prime indices of n.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.

Table[If[n==1,0,With[{q=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[q]-q[[1]]*Length[q]]],{n,100}]

a(n) = A056239(n) - A001222(n) * A055396(n).

a(n) = A056239(n) - A359360(n).

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]&]

A358169 Row n lists the first differences plus one of the prime indices of n with 1 prepended.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 1, 1, 3, 2, 3, 1, 5, 9, 1, 1, 1, 2, 3, 1, 1, 6, 2, 1, 1, 1, 1, 4, 10, 1, 2, 2, 11, 1, 1, 1, 1, 1, 2, 4, 1, 7, 3, 2, 1, 1, 2, 1, 12, 1, 8, 2, 5, 1, 1, 1, 3

Offset: 2

Gus Wiseman, Nov 01 2022

Every nonempty composition appears as a row exactly once.
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. Here this multiset is regarded as a sequence in weakly increasing order.
Also the reversed augmented differences of the integer partition with Heinz number n, where the augmented differences aug(q) of a sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k, and the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The non-reversed version is A355534.

			Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 1 2
   7: 4
   8: 1 1 1
   9: 2 1
  10: 1 3
  11: 5
  12: 1 1 2
  13: 6
  14: 1 4
  15: 2 2
  16: 1 1 1 1
  17: 7
  18: 1 2 1
  19: 8
  20: 1 1 3

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

Row-lengths are A001222.
The first term of each row is A055396.
Row-sums are A252464.
The rows appear to be ranked by A253566.
Another variation is A287352.
Constant rows have indices A307824.
The Heinz numbers of the rows are A325351.
Strict rows have indices A325366.
Row-minima are A355531, also A355524 and A355525.
Row-maxima are A355532, non-augmented A286470, also A355526.
Reversing rows gives A355534.
The non-augmented version A355536, also A355533.
A112798 lists prime indices, sum A056239.
Cf. A124010, A243055, A243056, A325352, A325390, A355523, A355528.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[primeMS[n],1]]+1,{n,30}]

A325555 Number of necklace compositions of n with distinct differences up to sign.

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 10, 15, 19, 24, 39, 49, 78, 106, 155, 207, 313, 430, 608, 867, 1239, 1670, 2313, 3220, 4483

Offset: 1

Gus Wiseman, May 11 2019

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

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) = 15 necklace compositions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)
       (11)  (12)  (13)   (14)   (15)   (16)    (17)
                   (22)   (23)   (24)   (25)    (26)
                   (112)  (113)  (33)   (34)    (35)
                          (122)  (114)  (115)   (44)
                                 (132)  (124)   (116)
                                        (133)   (125)
                                        (142)   (134)
                                        (223)   (143)
                                        (1132)  (152)
                                                (224)
                                                (233)
                                                (1124)
                                                (1142)
                                                (1322)

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

Cf. A000079, A008965, A235998, A242882, A325325, A325352, A325404, A325468, A325552, A325554, A325556.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[#]]&&neckQ[#]&]],{n,15}]

cp's OEIS Frontend

A359360 Length times minimum part of the integer partition with Heinz number n. Least prime index of n times number of prime indices of n.

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A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

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A356958 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (b-a+1, ..., y-a+1, z-a+1).

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A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

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A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

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

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A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

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

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