A302593 -id:A302593 - cp's OEIS frontend

A322900 Number of integer partitions of n whose parts are all proper powers of the same number.

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 7, 2, 11, 2, 9, 5, 11, 2, 16, 2, 18, 6, 17, 2, 27, 3, 23, 6, 30, 2, 38, 2, 37, 8, 39, 5, 58, 2, 49, 10, 66, 2, 74, 2, 78, 14, 77, 2, 109, 3, 100, 12, 118, 2, 131, 6, 146, 15, 143, 2, 190, 2, 169, 20, 203, 6, 224, 2, 242, 18, 248

Offset: 0

Gus Wiseman, Dec 30 2018

nonn

Such a partition contains either no 1's or only 1's.
A proper power of n is a number n^k for some positive integer k.
Also integer partitions whose parts all have the same radical base (A052410).

			The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):
  (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
      (11) (111) (22)   (11111) (33)     (1111111) (44)       (333)
                 (1111)         (42)               (422)      (111111111)
                                (222)              (2222)
                                (111111)           (11111111)
.
  (A)          (B)           (C)            (D)             (E)
  (55)         (11111111111) (66)           (1111111111111) (77)
  (82)                       (84)                           (842)
  (442)                      (93)                           (4442)
  (4222)                     (444)                          (8222)
  (22222)                    (822)                          (44222)
  (1111111111)               (3333)                         (422222)
                             (4422)                         (2222222)
                             (42222)                        (11111111111111)
                             (222222)
                             (111111111111)

a(n) = A072721(n) + 1.

Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@#&]],{n,30}]

A322901 Numbers whose prime indices are all powers of the same number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

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

			The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (21), (4), (111), (22), (31), (5), (211), (6), (41), (1111), (7), (221), (8), (311), (42), (51), (9), (2111), (33), (61), (222), (411).

Cf. A001597, A018819, A052409, A052410, A056239, A072720, A072721, A302242, A302593, A322900, A322902, A322903.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Select[Range[100],SameQ@@radbase/@DeleteCases[primeMS[#],1]&]

A322902 Numbers whose prime indices are all proper powers of the same number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

A prime index of n is a number m such that prime(m) divides n.
A proper power of n is a number n^k for some positive integer k.
Also the union of A322903 and A000079.

			The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (6), (1111), (7), (8), (42), (9), (33), (222).

Cf. A001597, A018819, A023893, A023894, A052410, A056239, A072720, A072721, A302242, A302593, A322900, A322901, A322903.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Select[Range[100],SameQ@@radbase/@primeMS[#]&]

A322903 Odd numbers whose prime indices are all proper powers of the same number.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 57, 59, 61, 63, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 179, 181, 189, 191

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

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

A proper power of n is a number n^k for some positive integer k.

			The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9), (3,3), (2,2,2), (10), (11), (12), (13), (14), (15), (4,4), (16), (8,2), (17), (18), (4,2,2), (19), (20), (21), (22), (2,2,2,2).

Cf. A001597, A018819, A023894, A052410, A056239, A072720, A072721, A302593, A322900, A322901, A322902.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Select[Range[100],And[OddQ[#],SameQ@@radbase/@primeMS[#]]&]

A357139 Take the weakly increasing prime indices of each prime index of n, then concatenate.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 5, 1, 3, 4, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 6, 1, 1, 1, 1, 4, 3, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2

Offset: 1

Gus Wiseman, Sep 29 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:  1
   4:
   5:  2
   6:  1
   7:  1 1
   8:
   9:  1 1
  10:  2
  11:  3
  12:  1
  13:  1 2
For example, the weakly increasing prime indices of 105 are (2,3,4), with prime indices ((1),(2),(1,1)), so row 105 is (1,2,1,1).

Row lengths are A302242.
Positions of strict rows are A302505.
Positions of constant rows are A302593.
Row sums are A325033, products A325032.
The version for standard compositions is A357135, rank A357134.
A000961 lists prime powers.
A003963 multiples prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A007716, A058891, A109082, A275024, A302243, A324926, A325034.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Join@@primeMS/@primeMS[n],{n,100}]

A322912 Number of integer partitions of n whose parts are all powers of the same squarefree number.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 33, 34, 42, 46, 56, 57, 71, 72, 88, 93, 109, 110, 134, 136, 158, 163, 191, 192, 229, 230, 266, 273, 311, 315, 370, 371, 419, 428, 491, 492, 565, 566, 642, 654, 730, 731, 836, 838, 936

Offset: 0

Gus Wiseman, Dec 30 2018

nonn

First differs from A072720 at a(12) = 33, A072720(12) = 34.

			The a(1) = 1 through a(8) = 15 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (71)
                    (211)   (311)    (51)      (421)      (422)
                    (1111)  (2111)   (222)     (511)      (611)
                            (11111)  (411)     (2221)     (2222)
                                     (2211)    (4111)     (3311)
                                     (3111)    (22111)    (4211)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Cf. A000961, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A302593, A322847, A322900, A322901, A322911.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n],And[And@@powsqfQ/@#,SameQ@@radbase/@DeleteCases[#,1]]&]],{n,30}]

A322911 Numbers whose prime indices are all powers of the same squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 62, 63, 64, 67, 68, 72, 73, 76, 79, 80, 81, 82, 83, 84, 86, 88, 92

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

The complement is {15, 30, 33, 35, 37, 39, 45, ...}. First differs from A318991 at a(33) = 38, A318991(33) = 37.

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The sequence lists all MM-numbers of multiset multisystems whose dual is constant, i.e. of the form {x,x,x,...,x} for some multiset x.

			The prime indices of 756 are {1,1,2,2,2,4}, which are all powers of 2, so 756 belongs to the sequence.
The prime indices of 841 are {10,10}, which are all powers of 10, so 841 belongs to the sequence.
The prime indices of 2645 are {3,9,9}, which are all powers of 3, so 2645 belongs to the sequence.
The prime indices of 3178 are {1,4,49}, which are all powers of squarefree numbers but not of the same squarefree number, so 3178 does not belong to the sequence.
The prime indices of 30599 are {12,144}, which are all powers of the same number 12, but this number is not squarefree, so 30599 does not belong to the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (6,2), (3,2,2), (7,2), (5,3), (3,2,1,1), (6,3), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (6,2,1), (7,3), (10,2), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).

Cf. A000688, A000961, A001597, A005117, A023893, A052410, A056239, A072720, A072774, A302242, A302593, A318400, A322847, A322901, A322912.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
sqfker[n_]:=Times@@First/@FactorInteger[n];
Select[Range[100],And[And@@powsqfQ/@primeMS[#],SameQ@@sqfker/@DeleteCases[primeMS[#],1]]&]

cp's OEIS Frontend

A322900 Number of integer partitions of n whose parts are all proper powers of the same number.

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A322901 Numbers whose prime indices are all powers of the same number.

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A322902 Numbers whose prime indices are all proper powers of the same number.

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A322903 Odd numbers whose prime indices are all proper powers of the same number.

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A357139 Take the weakly increasing prime indices of each prime index of n, then concatenate.

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A322912 Number of integer partitions of n whose parts are all powers of the same squarefree number.

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A322911 Numbers whose prime indices are all powers of the same squarefree number.

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