A322901 -id:A322901

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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).

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..56.

FORMULA

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

EXAMPLE

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)

MATHEMATICA

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 *)

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2002

STATUS

approved

A322900

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..70.

EXAMPLE

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)

MATHEMATICA

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All, 2]]);

Table[Length[Select[IntegerPartitions[n], SameQ@@radbase/@#&]], {n, 30}]

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 30 2018

STATUS

approved

A322902

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

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).

MATHEMATICA

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[#]&]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 30 2018

STATUS

approved

A322903

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

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).

MATHEMATICA

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[#]]&]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 30 2018

STATUS

approved

A322912

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

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)

MATHEMATICA

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 30 2018

STATUS

approved

A322911

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

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).

MATHEMATICA

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]]&]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 30 2018

STATUS

approved

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