A322902 -id:A322902 - cp's OEIS frontend

A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 10, 1, 8, 4, 10, 1, 15, 1, 17, 5, 16, 1, 26, 2, 22, 5, 29, 1, 37, 1, 36, 7, 38, 4, 57, 1, 48, 9, 65, 1, 73, 1, 77, 13, 76, 1, 108, 2, 99, 11, 117, 1, 130, 5, 145, 14, 142, 1, 189, 1, 168, 19, 202, 5, 223, 1, 241, 17, 247, 1, 309, 1, 286, 24, 333, 4

Offset: 0

Henry Bottomley, Jul 05 2002

nonn

First differs from A322968 at a(12) = 10, A322968(12) = 9.

			a(5)=1 since the only partition without 1 as a part is 5 (a power of 5). a(6)=4 since 6 can be written as 6 (powers of 6), 3+3 (powers of 3) and 4+2 and 2+2+2 (both powers of 2).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (C)
            (22)       (33)        (44)    (333)  (55)          (66)
                       (42)        (422)          (82)          (84)
                       (222)       (2222)         (442)         (93)
                                                  (4222)        (444)
                                                  (22222)       (822)
                                                                (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
(End)
Compare above to the example section of A379957. - _Antti Karttunen_, Jan 23 2025

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions.

Cf. A018819, A023894, A052410, A072720, A072721, A102430, A175082, A322900, A322902, A322903, A322968.

Cf. also A379957.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],SameQ@@radbase/@#]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)

a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&!ispower(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025

seq(n)={Vec(1 + sum(d=2, n, if(!ispower(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025

a(n) = A072721(n)-A072721(n-1). a(p)=1 for p prime.
a(n) = A322900(n) - 1. - Gus Wiseman, Jan 01 2019
G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A175082(k)^j)). - Andrew Howroyd, Jan 23 2025
For n >= 1, a(n) >= A379957(n). - Antti Karttunen, Jan 23 2025

A326536 MM-numbers of multiset partitions where every part has the same average.

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, 145, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A322902 in having 145.

These are numbers where each prime index has the same average 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. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index 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 sequence of multiset partitions where every part has the same average, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A051293, A112798, A302242, A320324, A326512, A326515, A326520, A326533, A326534, A326535, A326537.

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

A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 34, 35, 43, 47, 57, 58, 73, 74, 91, 96, 112, 113, 139, 141, 163, 168, 197, 198, 235, 236, 272, 279, 317, 321, 378, 379, 427, 436, 501, 502, 575, 576, 653, 666, 742, 743, 851, 853, 952, 963, 1080, 1081, 1211, 1216, 1361

Offset: 0

Henry Bottomley, Jul 05 2002

nonn

First differs from A322912 at a(12) = 34, A322912(12) = 33.

			a(6)=10 since 6 can be written as 6 (powers of 6), 5+1 (5), 4+1+1 (4 or 2), 3+3 (3), 3+1+1+1 (3), 4+2 (2), 2+2+2 (2), 2+2+1+1 (2), 2+1+1+1+1 (2) and 1+1+1+1+1+1 (powers of anything).
From _Gus Wiseman_, Jan 01 2019: (Start)
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)
(End)

Cf. A000123, A005704, A005705, A005706, A072721.

Cf. A018819, A023893, A052410, A102430, A322900, A322901, A322902, A322903, A322912.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@DeleteCases[#,1]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)

a(n) = a(n-1) + A072721(n). a(p) = a(p-1)+1 for p prime.

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

Original entry on oeis.org

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]&]

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[#]]&]

cp's OEIS Frontend

A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).

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A326536 MM-numbers of multiset partitions where every part has the same average.

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A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

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

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