A383014 -id:A383014 - cp's OEIS frontend

A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

12, 40, 63, 112, 144, 325, 351, 352, 675, 832, 931, 1008, 1539, 1600, 1728, 2176, 2875, 3509, 3969, 4864, 6253, 7047, 7056, 8775, 9072, 11776, 12427, 12544, 12691, 16128, 19133, 20736, 20800, 22464, 23125, 26973, 29403, 29696, 32269, 43200, 49392, 57967, 59711

Offset: 1

Gus Wiseman, Apr 14 2025

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, sum A056239.

All terms appear to have even sum of prime indices.

			The terms together with their prime indices begin:
     12: {1,1,2}
     40: {1,1,1,3}
     63: {2,2,4}
    112: {1,1,1,1,4}
    144: {1,1,1,1,2,2}
    325: {3,3,6}
    351: {2,2,2,6}
    352: {1,1,1,1,1,5}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
    931: {4,4,8}
   1008: {1,1,1,1,2,2,4}
   1539: {2,2,2,2,8}
   1600: {1,1,1,1,1,1,3,3}
   1728: {1,1,1,1,1,1,2,2,2}

Compositions of this type are counted by A353851, ranked by A353848.
Positions of terms > 1 in A382877, zeros A383100 (complement A383014).
For run-lengths instead of sums we have A383089, counted by A383090.
The complement for run-lengths instead of sums is A383091, counted by A383092
Partitions of this type are counted by A383097.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A000961, A001221, A001222, A329738, A353833, A354584, A381636, A381871, A382857, A382876, A382879.

Select[Range[100],Length[Select[Permutations[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]],SameQ@@Total/@Split[#]&]]>1&]

A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 2, 2, 0, 1, 1, 1, 0, 0, 3, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 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, Mar 19 2025

nonn

Also the number of factorizations of n into prime powers > 1 with equal sums of prime indices.

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 2 multiset partitions into constant blocks with a common sum:
  {{2,2},{1,1,1,1}}
  {{2},{2},{1,1},{1,1}}
so a(144) = 2.

For just constant blocks we have A000688.
Twice-partitions of this type are counted by A279789.
For just a common sum we have A321455.
For distinct instead of equal sums we have A381635.
Positions of 0 are A381871, counted by A381993.
MM-numbers of these multiset partitions are A382215.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A279784, A295935, A381453 (lower), A381455 (upper).
Cf. A000720, A000961, A001222, A006171, A265947, A381633, A381719.
Cf. A323774, A382076, A382204, A382524, A383014, A383093, A383309.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[mps[prix[n]], SameQ@@Total/@#&&And@@SameQ@@@#&]],{n,100}]

A323774(n) = Sum_{A056239(k)=n} a(k). Gus Wiseman, Apr 25 2025

A383089 Numbers whose prime indices have more than one permutation with all equal run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Apr 18 2025

nonn

First differs from A362606 (complement A359178 with 1) in having 180 and lacking 240.
First differs from A130092 (complement A130091) in having 360 and lacking 240.
First differs from A351295 (complement A351294) in having 216 and lacking 240.
Includes all squarefree numbers A005117 except the primes A000040.
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.

			The prime indices of 36 are {1,1,2,2}, and we have 4 permutations each having all equal run-lengths: (1,1,2,2), (1,2,1,2), (2,2,1,1), (2,1,2,1), so 36 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   46: {1,9}
   51: {2,7}
   55: {3,5}
   57: {2,8}
   58: {1,10}
   60: {1,1,2,3}

Positions of terms > 1 in A382857 (distinct A382771), zeros A382879, ones A383112.
For run-sums instead of lengths we have A383015, counted by A383097.
Partitions of this type are counted by A383090.
The complement is A383091, counted by A383092, just zero A382915, just one A383094.
For distinct instead of equal run-sums we have A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A047966 counts partitions with equal run-lengths, compositions A329738.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A000961, A001221, A001222, A048767, A353744, A353833, A381541, A381871, A382877, A383014, A383100.

Select[Range[100],Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]>1&]

The complement is A383091 = A382879 \/ A383112, counted by A382915 + A383094.

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 9, 5, 9, 2, 23, 2, 11, 10, 24, 2, 33, 2, 36, 12, 15, 2, 87, 7, 17, 17, 53, 2, 96, 2, 79, 16, 21, 14, 196, 2, 23, 18, 154, 2, 166, 2, 99, 54, 27, 2, 431, 9, 85, 22, 128, 2, 303, 18, 261, 24, 33, 2, 771, 2, 35, 73, 331, 20, 422, 2, 198, 28, 216, 2, 1369

Offset: 0

Gus Wiseman, Apr 22 2025

nonn

			The partition (4,4,2,2,2,2,1,1,1,1,1,1,1,1) has two partitions into constant blocks with a common sum: {{4,4},{2,2,2,2},{1,1,1,1,1,1,1,1}} and {{4},{4},{2,2},{2,2},{1,1,1,1},{1,1,1,1}}, so is counted under a(24).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (2211)               (2222)
                                     (3111)               (22211)
                                     (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)

Twice-partitions of this type (constant with common) are counted by A279789.
Multiset partitions of this type are ranked by A383309.
The complement is counted by A381993, ranks A381871.
For sets we have the complement of A381994, see A381719, A382080.
Normal multiset partitions of this type are counted by A382203, sets A381718.
For distinct instead of equal block-sums we have A382427.
These partitions are ranked by A383014, nonzeros of A381995.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers, see A381715.
A323774 counts partitions into constant blocks with a common sum
Constant blocks with distinct sums: A381635, A381636, A381717.
Permutation with equal run-sums: A383096, A383098, A383100, A383110
Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]>0&]],{n,0,30}]

Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

More terms from Jakub Buczak, May 03 2025

A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 10, 13, 15, 13, 31

Offset: 0

Gus Wiseman, Apr 25 2025

Any strict partition can be partitioned into a single set, so we have a lower bound a(n) >= A000009(n).

			The multiset (3,2,2,1,1) has partition {{3},{1,2},{1,2}}, so is counted under a(9).
The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)        (54)
             (111)  (31)    (41)     (42)      (52)       (53)        (63)
                    (1111)  (11111)  (51)      (61)       (62)        (72)
                                     (222)     (421)      (71)        (81)
                                     (321)     (1111111)  (431)       (333)
                                     (2211)               (521)       (432)
                                     (111111)             (2222)      (531)
                                                          (3311)      (621)
                                                          (11111111)  (3321)
                                                                      (32211)
                                                                      (222111)
                                                                      (111111111)

Twice-partitions of this type (into sets with a common sum) are counted by A279788.
Multiset partitions of this type are ranked by A326534 /\ A302478.
For distinct instead of equal sums we have A381992, see also A382077.
The complement is counted by A381994, ranks A381719.
Partitions of prime indices of this type are counted by A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
For constant instead of strict blocks we have A383093, ranks A383014.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A050320, A293511, A321455, A358914, A381633, A381717, A381993.

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

A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 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, 100, 102, 104, 105

Offset: 1

Gus Wiseman, Apr 17 2025

nonn

First differs from A381871 in having 36.

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.

			The prime indices of 36 are {1,1,2,2}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}

For run-lengths instead of sums we have A059404, distinct A130092.
The complement is A353833, counted by A304442.
For distinct instead of equal run-sums we have A353839.
Partitions of this type are counted by A382076.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with a common run-sum, ranks A353848.
A353862 gives the greatest run-sum of prime indices, least A353931.
A382877 counts permutations of prime indices with equal run-sums, zeros A383100.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.

Select[Range[100], !SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

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A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

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A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

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A383089 Numbers whose prime indices have more than one permutation with all equal run-lengths.

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A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

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A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

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A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

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