A323306 -id:A323306 - cp's OEIS frontend

A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

1, 1, 3, 3, 6, 3, 11, 3, 12, 6, 13, 3, 52, 3, 15, 30, 57, 3, 156, 3, 238, 129, 19, 3, 2221, 6, 21, 415, 3114, 3, 14921, 3, 12853, 1044, 25, 6219, 164743, 3, 27, 2220, 851476, 3, 954088, 3, 434106, 3326714, 31, 3, 24648724, 6, 22309800, 7269, 2737618, 3, 69823653

Offset: 0

Gus Wiseman, Jan 13 2019

nonn

Also the number of non-normal semi-magic rectangles summing to n with no zeros.

Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - Chai Wah Wu, Jan 13 2019

			The a(6) = 11 matrices:
  [6] [3 3] [2 2 2] [1 1 1 1 1 1]
.
  [3] [1 2] [2 1] [1 1 1]
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1]
  [2] [1 1]
  [2] [1 1]
.
  [1]
  [1]
  [1]
  [1]
  [1]
  [1]

Chai Wah Wu, Table of n, a(n) for n = 0..59

Cf. A000219, A006052, A120733, A305551, A319056, A321719, A323295, A323300, A323302, A323306, A323347, A323349.

Table[Length[Select[Join@@Table[Partition[cmp,d],{cmp,Join@@Permutations/@IntegerPartitions[n]},{d,Divisors[Length[cmp]]}],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,10}]

a(p) = 3 and a(p^2) = 6 for p prime (see comment). - Chai Wah Wu, Jan 13 2019

a(21)-a(31) from Chai Wah Wu, Jan 13 2019
a(32)-a(53) from Chai Wah Wu, Jan 14 2019
a(54) from Chai Wah Wu, Jan 16 2019

A323347 Number of integer partitions of n whose parts can be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 6, 2, 11, 2, 7, 7, 10, 2, 18, 2, 17, 13, 9, 2, 50, 3, 10, 24, 34, 2, 85, 2, 51, 46, 12, 9, 261, 2, 13, 80, 257, 2, 258, 2, 323, 431, 15, 2, 1533, 3, 227, 206, 1165, 2, 971, 483, 2409, 309, 18, 2

Offset: 0

Gus Wiseman, Jan 13 2019

Rectangles must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 2 for p prime, since the only allowable rectangles must be of size 1 X 1 corresponding to the partition (p), or 1 X p or p X 1 corresponding to the partition (1,1,...,1). Similarly, a(p^2) = 3 since the allowable rectangles must be of sizes 1 X 1 (partition (p^2)), 1 X p or p X 1 (partition (p,p,...,p)), 1 X p^2, p^2 X 1 and p X p (partition (1,1,...,1)). - Chai Wah Wu, Jan 14 2019

			The a(8) = 5 integer partitions are (8), (44), (2222), (3311), (11111111).
The a(12) = 11 integer partitions (C = 12):
  (C)
  (66)
  (444)
  (3333)
  (4422)
  (5511)
  (222222)
  (332211)
  (22221111)
  (222111111)
  (111111111111)
For example, the arrangements of (222111111) are:
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

A000041(n) = A323348(n) + a(n).
Cf. A006052, A007016, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323304, A323306, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[IntegerPartitions[n],!Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]

a(p) = 2 and a(p^2) = 3 for p prime (see comment). - Chai Wah Wu, Jan 14 2019

a(17)-a(53) from Chai Wah Wu, Jan 15 2019

a(54)-a(59) from Chai Wah Wu, Jan 16 2019

A323302 Number of ways to arrange the parts of the integer partition with Heinz number n into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 0, 3, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

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

			The a(900) = 12 matrix-arrangements of (3,3,2,2,1,1):
  [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
  [3 2 1] [3 1 2] [2 3 1] [2 1 3] [1 3 2] [1 2 3]
.
  [1 3] [1 3] [2 2] [2 2] [3 1] [3 1]
  [2 2] [3 1] [1 3] [3 1] [1 3] [2 2]
  [3 1] [2 2] [3 1] [1 3] [2 2] [1 3]

Positions of zeros are a superset of A106543.
Cf. A000005, A001222, A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
Cf. A323300, A323303, A323305, A323306, A323347, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[ptnmats[n],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,100}]

A325247 Numbers whose omega-sequence is strict (no repeated parts).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

First differs from A323306 in having 216.
We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).
Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
     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}
    23: {9}
    25: {3,3}
    27: {2,2,2}
    29: {10}
    31: {11}
    32: {1,1,1,1,1}
    36: {1,1,2,2}

Positions of squarefree numbers in A325248.
Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.
Omega-sequence statistics: A001221 (second omega), A001222 (first omega), A071625 (third omega), A304465 (second-to-last omega), A182850 or A323014 (depth), A323022 (fourth omega), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#1]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],UnsameQ@@omseq[#]&]

A323304 Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

The first term of this sequence absent from A106543 is 144.

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

Index entries for sequences related to Heinz numbers

Complement of A323306.
Cf. A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
Cf. A323300, A323302, A323305, A323347, A323348,  A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[2,1000],Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 27, 36, 54, 66, 99, 128, 169, 221, 295, 367, 488, 610, 779, 993, 1253, 1525, 1955, 2426, 2986, 3684, 4563, 5519, 6840, 8298, 10097, 12298, 14874, 17716, 21635, 26002, 31105, 37081, 44581, 52916, 63259, 74852, 88703, 105543, 124752, 145740, 173522, 203999, 239737, 280424, 329929

Offset: 0

Gus Wiseman, Jan 13 2019

nonn

			The a(8) = 17 integer partitions:
  (53), (62), (71),
  (332), (422), (431), (521), (611),
  (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111).

Chai Wah Wu, Table of n, a(n) for n = 0..59

A000041(n) = A323347(n) + a(n).
Cf. A006052, A007016, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323304, A323306, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[IntegerPartitions[n],Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]

a(17)-a(53) from Chai Wah Wu, Jan 15 2019

A323303 Number of ways to arrange the prime indices of n into a matrix with equal column-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 6, 1, 2, 2, 2, 2, 10, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 12, 1, 2, 3, 4, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 3, 3, 2, 6, 1, 5, 3, 2, 1, 12, 2, 2

Offset: 1

Gus Wiseman, Jan 17 2019

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.

			The a(90) = 16 matrix-arrangements of (3,2,2,1) with equal column-sums:
  [1 2] [2 1] [2 3] [3 2]
  [3 2] [2 3] [2 1] [1 2]
.
  [1] [1] [1] [2] [2] [2] [2] [2] [2] [3] [3] [3]
  [2] [2] [3] [1] [1] [2] [2] [3] [3] [1] [2] [2]
  [2] [3] [2] [2] [3] [1] [3] [1] [2] [2] [1] [2]
  [3] [2] [2] [3] [2] [3] [1] [2] [1] [2] [2] [1]

Cf. A006052, A008480, A056239, A112798, A118914, A120733, A319056, A321719.

Cf. A323295, A323300, A323302, A323305, A323306, A323347, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[ptnmats[n],SameQ@@Total/@Transpose[#]&]],{n,100}]

A323524 Number of integer partitions of n whose parts can be arranged into a square matrix with equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 4, 6, 1, 10, 1, 7, 10, 6, 1, 24, 2, 7, 22, 18, 1, 38, 1, 35, 43, 9, 6, 124, 1, 10, 77, 158, 1, 110, 1, 285, 186, 12, 1, 742, 2, 170, 203, 1110, 1, 285, 480, 2115, 306, 15, 1

Offset: 0

Gus Wiseman, Jan 17 2019

			The a(12) = 5 integer partitions are (12), (5,5,1,1), (4,4,2,2), (3,3,3,3), (2,2,2,1,1,1,1,1,1). For example, such a matrix for (2,2,2,1,1,1,1,1,1) is:
  [1 1 2]
  [2 1 1]
  [1 2 1]

Cf. A006052, A007016, A103198, A120732, A321719, A321722, A321725, A323306, A323347, A323349, A323523.

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

a(16)-a(59) from Chai Wah Wu, Jan 20 2019

A359390 Sequence lists the numbers k such that bottom entry is an integer in the ratio d(i+1)/d(i) triangle of the elements in the divisors of n, where d(1) < d(2) < ... < d(q) denote the divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Michel Lagneau, Jan 03 2023

nonn

The corresponding integer bottom entry is 1 if k is nonprime or k if k is prime. [It is very likely that this is true, but no proof has yet been given. - Jianing Song, Jan 22 2023]
We observe that a(n) = A323306(n) for n = 1..50. But a(51) = 144 does not belong to that sequence.
Note that the bottom rational is Product_{i=1..q} d(i) ^ (binomial(q-1,i-1) * (-1)^(q-i)). - Kevin Ryde, Jan 03 2023
Given n, let 1 = M(1,1) < M(1,2) < ... < M(1,d) = n be the divisors of n, and M(i,j) = M(i-1,j+1)/M(i-1,j) for 2 <= i <= d, 1 <= j <= d+1-i. Since M(1,d+1-j) = n/M(1,j) for 1 <= j <= d, we have M(i,d+2-i-j) = M(i,j) for even i, 1 <= j <= d+1-i, and M(i,d+2-i-j) = 1/M(i,j) for odd i > 1, 1 <= j <= d+1-i. If n is a square, then d is odd, so M(d,1) = 1/M(d,1) => M(d,1) = 1. This shows that all square numbers are terms. Note that all powers of primes (A000961) are trivially terms. It seems that the squares and the powers of primes are the only terms. - Jianing Song, Jan 03 2023

			36 is a term because the triangle of the elements d(i+1)/d(i) has bottom entry 1:
  [1, 2, 3, 4, 6, 9, 12, 18, 36]
  [2, 3/2, 4/3, 3/2, 3/2, 4/3, 3/2, 2]
  [3/4, 8/9, 9/8, 1, 8/9, 9/8, 4/3]
  [32/27, 81/64, 8/9, 8/9, 81/64, 32/27]
  [2187/2048, 512/729, 1, 729/512, 2048/2187]
  [1048576/1594323, 729/512, 729/512, 1048576/1594323]
  [1162261467/536870912, 1, 536870912/1162261467]
  [536870912/1162261467, 536870912/1162261467]
  [1].
6 is not a term because the triangle of the elements d(i+1)/d(i) has bottom entry 16/9.
  [1, 2, 3, 6]
  [2, 3/2, 2]
  [3/4, 4/3]
  [16/9]

Cf. A323306. Contains A000290 and A000961 as subsequences (and conjectured to be the union of these two sequences).

Lst={}; Table[d=Divisors[n]; While[Length[d]>1,d=Ratios[d]]; If[d[[1]]==Floor[d[[1]]],AppendTo[Lst,n]],{n,300}]; Lst

isA359390(n) = my(L = factor(n), w = #L~, v=divisors(n), q=#v); for(i_d=1, q-1, for(i_p=1, w, L[i_p,2] += binomial(q-1,i_d-1) * (-1)^(q-i_d) * valuation(v[i_d], L[i_p,1]))); for(i_p=1, w, if(L[i_p,2]<0, return(0))); return(1) \\ Jianing Song, Jan 22 2023, based on the formula provided by Kevin Ryde

cp's OEIS Frontend

A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

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A323347 Number of integer partitions of n whose parts can be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

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A323302 Number of ways to arrange the parts of the integer partition with Heinz number n into a matrix with equal row-sums and equal column-sums.

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A325247 Numbers whose omega-sequence is strict (no repeated parts).

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A323304 Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.

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A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

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A323303 Number of ways to arrange the prime indices of n into a matrix with equal column-sums.

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A323524 Number of integer partitions of n whose parts can be arranged into a square matrix with equal row and column sums.

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A359390 Sequence lists the numbers k such that bottom entry is an integer in the ratio d(i+1)/d(i) triangle of the elements in the divisors of n, where d(1) < d(2) < ... < d(q) denote the divisors of k.

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