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.

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A336131 Number of ways to split an integer partition of n into contiguous subsequences all having different sums.

Original entry on oeis.org

1, 1, 2, 6, 9, 20, 44, 74, 123, 231, 441, 681, 1188, 1889, 3110, 5448, 8310, 13046
Offset: 0

Views

Author

Gus Wiseman, Jul 11 2020

Keywords

Examples

			The a(1) = 1 through a(4) = 9 splits:
  (1)  (2)    (3)        (4)
       (1,1)  (2,1)      (2,2)
              (1,1,1)    (3,1)
              (2),(1)    (2,1,1)
              (1),(1,1)  (3),(1)
              (1,1),(1)  (1,1,1,1)
                         (2,1),(1)
                         (1),(1,1,1)
                         (1,1,1),(1)

Links

Crossrefs

The version with equal instead of different sums is A317715.
Starting with a composition gives A336127.
Starting with a strict composition gives A336128.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A323433, A336130, A336134, A336135.

Programs

  • Mathematica
    splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277
Offset: 1

Views

Author

Gus Wiseman, Jul 25 2022

Keywords

Examples

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

Crossrefs

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Formula

Intersection of A005117 and A355743.

A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 27, 37, 62, 82, 125, 168, 246, 320, 462, 585, 839, 1078, 1466, 1830, 2528, 3136, 4188, 5210, 6907, 8498, 11177, 13570, 17668, 21614, 27580, 33339, 42817, 51469, 65083, 78457, 98409, 117602, 147106, 174663, 217400, 259318, 319076, 377707
Offset: 0

Views

Author

Gus Wiseman, Jul 11 2020

Keywords

Examples

			The a(1) = 1 through a(6) = 17 splits:
  (1)  (2)    (3)        (4)          (5)            (6)
       (1,1)  (2,1)      (2,2)        (3,2)          (3,3)
              (1,1,1)    (3,1)        (4,1)          (4,2)
              (1),(1,1)  (2,1,1)      (2,2,1)        (5,1)
                         (1,1,1,1)    (3,1,1)        (2,2,2)
                         (1),(1,1,1)  (2,1,1,1)      (3,2,1)
                                      (2),(2,1)      (4,1,1)
                                      (1,1,1,1,1)    (2,2,1,1)
                                      (2),(1,1,1)    (2),(2,2)
                                      (1),(1,1,1,1)  (3,1,1,1)
                                      (1,1),(1,1,1)  (2,1,1,1,1)
                                                     (2),(2,1,1)
                                                     (1,1,1,1,1,1)
                                                     (2),(1,1,1,1)
                                                     (1),(1,1,1,1,1)
                                                     (1,1),(1,1,1,1)
                                                     (1),(1,1),(1,1,1)

Links

Crossrefs

The version with equal sums is A317715.
The version with strictly decreasing sums is A336135.
The version with weakly decreasing sums is A316245.
The version with different sums is A336131.
Starting with a composition gives A304961.
Starting with a strict partition gives A336133.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A318684, A323433, A336128, A336130.

Programs

  • Mathematica
    splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]
  • PARI
    a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r > t && t >= s, self()(r,m,t+1,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m), s,t+m,1))); recurse(n,n,0,0,0)} \\ Andrew Howroyd, Jan 18 2024

Extensions

a(21) onwards from Andrew Howroyd, Jan 18 2024

A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 37, 76, 159, 326, 671, 1376, 2815, 5759, 11774, 24083, 49249, 100632, 205490, 419420, 855799, 1745889, 3561867, 7268240, 14836127, 30295633, 61888616
Offset: 0

Views

Author

Gus Wiseman, Mar 26 2025

Keywords

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

Examples

			The a(1) = 1 through a(4) = 9 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}
                                   {{2},{1,1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{1},{2},{3},{4}}
The a(5) = 19 factorizations:
  32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
      4*8   2*4*9    2*3*5*9
      2*81  2*3*8    2*3*5*49
      4*27  2*3*125  2*3*7*25
      9*8   2*9*25
      3*16  2*5*27
            5*4*9

Crossrefs

Without distinct sums we have A055887.
Twice-partitions of this type are counted by A279786.
For distinct blocks instead of sums we have A304969.
Without constant blocks we have A326519.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A381718.
For equal instead of distinct block-sums we have A382204.
For equal block-sums and strict blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A116540, A255906, A317532.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

Extensions

a(14)-a(26) from Christian Sievers, Apr 04 2025

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79
Offset: 1

Views

Author

Gus Wiseman, Mar 22 2025

Keywords

Comments

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.
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, sum A056239.

Examples

			The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

Crossrefs

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

Programs

  • Mathematica
    hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 17, 27, 43, 46, 82, 103, 133, 181, 258, 295
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2025

Keywords

Examples

			For y = (3,3,1,1) we have {{1,3},{1,3}}, so y is not counted under a(8).
For y = (3,2,2,1), although we have {{1,3},{2,2}}, the block {2,2} is not a set, so y is counted under a(8).
The a(4) = 1 through a(8) = 12 partitions:
  (2,1,1)  (2,2,1)    (4,1,1)      (3,2,2)        (3,3,2)
           (3,1,1)    (3,1,1,1)    (3,3,1)        (4,2,2)
           (2,1,1,1)  (2,1,1,1,1)  (5,1,1)        (6,1,1)
                                   (2,2,2,1)      (3,2,2,1)
                                   (3,2,1,1)      (4,2,1,1)
                                   (4,1,1,1)      (5,1,1,1)
                                   (2,2,1,1,1)    (2,2,2,1,1)
                                   (3,1,1,1,1)    (3,2,1,1,1)
                                   (2,1,1,1,1,1)  (4,1,1,1,1)
                                                  (2,2,1,1,1,1)
                                                  (3,1,1,1,1,1)
                                                  (2,1,1,1,1,1,1)

Crossrefs

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279788.
Interchanging "constant" with "strict" gives A381717, see A381635, A381636, A381991.
Normal multiset partitions of this type are counted by A381718, see A279785.
These partitions are ranked by A381719, zeros of A382080.
For distinct instead of equal block-sums we have A381990, ranked by A381806.
For constant instead of strict blocks we have A381993.
A000041 counts integer partitions, strict A000009.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A381633 counts set systems with distinct sums, see A381634, A293243.
Cf. A002846, A047966, A279786, A279789, A293511, A299202, A317142, A358914, A381992.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#], And@@UnsameQ@@@#&&SameQ@@Total/@#&]]==0&]],{n,0,10}]

A296131 Number of twice-factorizations of n where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 9, 2, 2, 2, 9, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 13, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 11, 1, 2, 4, 16, 2, 5, 1, 4, 2, 5, 1, 18, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 11, 2
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Comments

a(n) is the number of ways to choose a perfect divisor of each factor in a strict factorization of n.

Examples

			The a(24) = 8 twice-factorizations: (2)*(3)*(2*2), (2)*(3)*(4), (2)*(12), (3)*(2*2*2), (3)*(8), (2*2)*(6), (4)*(6), (24).

Crossrefs

Cf. A001055, A005117, A045778, A063834, A089723, A279786, A281113, A296120, A296118.

Programs

  • Mathematica
    sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Product[DivisorSigma[0,GCD@@FactorInteger[d][[All,2]]],{d,fac}],{fac,sfs[n]}],{n,100}]

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A089723(n)/n^s).

A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720
Offset: 0

Views

Author

Gus Wiseman, Jul 11 2020

Keywords

Examples

			The a(1) = 1 through a(9) = 9 splittings:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
                                 (3,2,1)  (6,1)    (7,1)    (7,2)
                                          (4,2,1)  (4,3,1)  (8,1)
                                                   (5,2,1)  (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
                                                            (4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.

Links

Crossrefs

The version with equal sums is A318683.
The version with strictly decreasing sums is A318684.
The version with weakly decreasing sums is A319794.
The version with different sums is A336132.
Starting with a composition gives A304961.
Starting with a non-strict partition gives A336134.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.

Programs

  • Mathematica
    splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 31, 40, 73, 98, 158, 204, 340, 420, 629, 819, 1202, 1494, 2174, 2665, 3759, 4688, 6349, 7806, 10788, 13035, 17244, 21128, 27750, 33499, 43941, 52627, 67957, 81773, 103658, 124047, 158628, 187788, 235162, 280188, 349612, 413120, 513952, 604568
Offset: 0

Views

Author

Gus Wiseman, Jul 11 2020

Keywords

Examples

			The a(1) = 1 through a(5) = 15 splittings:
  (1)  (2)      (3)          (4)              (5)
       (1,1)    (2,1)        (2,2)            (3,2)
       (1),(1)  (1,1,1)      (3,1)            (4,1)
                (1),(1,1)    (2,1,1)          (2,2,1)
                (1),(1),(1)  (2),(2)          (3,1,1)
                             (1,1,1,1)        (2,1,1,1)
                             (2),(1,1)        (2),(2,1)
                             (1),(1,1,1)      (1,1,1,1,1)
                             (1,1),(1,1)      (2),(1,1,1)
                             (1),(1),(1,1)    (1),(1,1,1,1)
                             (1),(1),(1),(1)  (1,1),(1,1,1)
                                              (1),(1),(1,1,1)
                                              (1),(1,1),(1,1)
                                              (1),(1),(1),(1,1)
                                              (1),(1),(1),(1),(1)

Links

Crossrefs

The version with weakly decreasing sums is A316245.
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with strictly decreasing sums is A336135.
The version with different sums is A336131.
Starting with a composition gives A075900.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.

Programs

  • Mathematica
    splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]
  • PARI
    a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ Andrew Howroyd, Jan 18 2024

Extensions

a(21) onwards from Andrew Howroyd, Jan 18 2024

A302496 Products of distinct primes of prime-power index.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 70, 77, 82, 83, 85, 93, 95, 97, 102, 103, 105, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159
Offset: 1

Views

Author

Gus Wiseman, Apr 09 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of constant-multiset systems.
01: {}
02: {{}}
03: {{1}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
10: {{},{2}}
11: {{3}}
14: {{},{1,1}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
35: {{2},{1,1}}
38: {{},{1,1,1}}

Crossrefs

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279786, A281113, A296131, A301767, A302242, A302243, A302493, A302494.

Programs

  • Mathematica
    Select[Range[nn],Or[#===1,SquareFreeQ[#]&&And@@PrimePowerQ/@PrimePi/@DeleteCases[FactorInteger[#][[All,1]],2]]&]
  • PARI
    is(n) = if(bigomega(n)!=omega(n), return(0), my(f=factor(n)[, 1]~); for(k=1, #f, if(!isprimepower(primepi(f[k])) && primepi(f[k])!=1, return(0)))); 1 \\ Felix Fröhlich, Apr 10 2018
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