cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A323300 Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 2, 2, 4, 1, 6, 1, 4, 4, 3, 1, 6, 1, 6, 4, 4, 1, 12, 2, 4, 2, 6, 1, 12, 1, 2, 4, 4, 4, 18, 1, 4, 4, 12, 1, 12, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 12, 4, 12, 4, 4, 1, 36, 1, 4, 6, 4, 4, 12, 1, 6, 4, 12, 1, 20, 1, 4, 6, 6, 4, 12, 1, 10, 3, 4
Offset: 1

Views

Author

Gus Wiseman, Jan 12 2019

Keywords

Comments

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

Examples

			The a(24) = 12 matrices whose entries are (2,1,1,1):
  [1 1 1 2] [1 1 2 1] [1 2 1 1] [2 1 1 1]
.
  [1 1] [1 1] [1 2] [2 1]
  [1 2] [2 1] [1 1] [1 1]
.
  [1] [1] [1] [2]
  [1] [1] [2] [1]
  [1] [2] [1] [1]
  [2] [1] [1] [1]

Crossrefs

Positions of 1's are one and prime numbers A008578.
Positions of 2's are primes to prime powers A053810.
Cf. A000005, A001222, A008480, A056239, A063989, A112798, A120733.
Cf. A323295, A323305, A323307, A323351.

Programs

  • Mathematica
    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/@#&];
Array[Length[ptnmats[#]]&,100]

Formula

a(n) = A008480(n) * A000005(A001222(n)).

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

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Comments

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

Examples

			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]

Crossrefs

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.

Programs

  • Mathematica
    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}]

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

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Mathematica
    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[#]]&]=={}&]

A323350 Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376
Offset: 1

Views

Author

Gus Wiseman, Jan 15 2019

Keywords

Comments

First differs from A014613 in having 512.

Examples

			360 = 2*2*2*3*3*5 has 6 prime factors, and 6 is not a perfect square, so 360 does not belong to the sequence.
2160 = 2*2*2*2*3*3*3*5 has 8 prime factors, and 8 is not a perfect square, so 2160 does not belong to the sequence.
10800 = 2*2*2*2*3*3*3*5*5 has 9 prime factors, and 9 is a perfect square, so 10800 belongs to the sequence.

Links

Crossrefs

Cf. A000005, A001222, A063989, A067028, A067340, A070175, A071625, A118914, A167175, A323305.

Programs

  • Maple
    filter:= proc(n) local t;
  t:= numtheory:-bigomega(n);
  t > 1 and issqr(t)
end proc:
select(filter, [$4..1000]); # Robert Israel, Jan 15 2019
  • Mathematica
    Select[Range[100],#>1&&!PrimeQ[#]&&IntegerQ[Sqrt[PrimeOmega[#]]]&]
  • PARI
    isok(n) = (n>1) && !isprime(n) && issquare(bigomega(n)); \\ Michel Marcus, 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

Views

Author

Gus Wiseman, Jan 17 2019

Keywords

Comments

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.

Examples

			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]

Crossrefs

Cf. A006052, A008480, A056239, A112798, A118914, A120733, A319056, A321719.
Cf. A323295, A323300, A323302, A323305, A323306, A323347, A323349.

Programs

  • Mathematica
    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}]
Showing 1-5 of 5 results.

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