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Search: a322902 -id:a322902
Displaying 1-6 of 6 results found. page 1
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A326536 MM-numbers of multiset partitions where every part has the same average. +10
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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}}.
LINKS
Table of n, a(n) for n=1..61.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.
EXAMPLE
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: {{},{},{},{},{}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], SameQ@@Mean/@primeMS/@primeMS[#]&]
CROSSREFS
Cf. A038041, A051293, A112798, A302242, A320324, A326512, A326515, A326520, A326533, A326534, A326535, A326537.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2019
STATUS
approved
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
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
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Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)