A322903 -id:A322903

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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

A072721

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

+10
8

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

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

OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..77.

FORMULA

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

a(n) = A322900(n) - 1. - Gus Wiseman, Jan 01 2019

EXAMPLE

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)

MATHEMATICA

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 *)

CROSSREFS

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

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

A322901

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

+10
6

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

(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.

LINKS

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

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), (21), (4), (111), (22), (31), (5), (211), (6), (41), (1111), (7), (221), (8), (311), (42), (51), (9), (2111), (33), (61), (222), (411).

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/@DeleteCases[primeMS[#], 1]&]

CROSSREFS

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

A322968

Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

+10
1

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 9, 1, 8, 4, 10, 1, 14, 1, 16, 5, 16, 1, 24, 2, 22, 5, 28, 1, 37, 1, 36, 7, 38, 4, 55, 1, 48, 9, 63, 1, 73, 1, 76, 12, 76, 1, 105, 2, 98, 11, 116, 1, 128, 5, 143, 14, 142, 1, 186, 1, 168, 18, 202, 5

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

OFFSET

0,5

COMMENTS

First differs from A072721 at a(12) = 9, A072721(12) = 10.

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):

(2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (66)

(22) (33) (44) (333) (55) (84)

(42) (422) (82) (93)

(222) (2222) (442) (444)

(4222) (822)

(22222) (3333)

(4422)

(42222)

(222222)

The a(20) = 16 integer partitions:

(10,10), (16,4),

(8,8,4), (16,2,2),

(5,5,5,5), (8,4,4,4), (8,8,2,2),

(4,4,4,4,4), (8,4,4,2,2),

(4,4,4,4,2,2), (8,4,2,2,2,2),

(4,4,4,2,2,2,2), (8,2,2,2,2,2,2),

(4,4,2,2,2,2,2,2),

(4,2,2,2,2,2,2,2,2),

(2,2,2,2,2,2,2,2,2,2).

MATHEMATICA

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

powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];

Table[Length[Select[IntegerPartitions[n], And[FreeQ[#, 1], And@@powsqfQ/@#, SameQ@@radbase/@#]&]], {n, 30}]

CROSSREFS

Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 01 2019

STATUS

approved

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