cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Henry Bottomley, Jul 05 2002

Keywords

Comments

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

Examples

			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

Links

Crossrefs

Cf. A018819, A023894, A052410, A072720, A072721, A102430, A175082, A322900, A322902, A322903, A322968.
Cf. also A379957.

Programs

  • 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 *)
  • PARI
    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
  • PARI
    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

Formula

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

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

Views

Author

Henry Bottomley, Jul 05 2002

Keywords

Comments

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

Examples

			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)

Crossrefs

Cf. A000123, A005704, A005705, A005706, A072721.
Cf. A018819, A023893, A052410, A102430, A322900, A322901, A322902, A322903, A322912.

Programs

  • 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 *)

Formula

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

Views

Author

Gus Wiseman, Dec 30 2018

Keywords

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).

Examples

			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)

Crossrefs

a(n) = A072721(n) + 1.
Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Dec 30 2018

Keywords

Comments

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

Examples

			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).

Crossrefs

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

Programs

  • 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]&]

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

Views

Author

Gus Wiseman, Dec 30 2018

Keywords

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.

Examples

			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).

Crossrefs

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

Programs

  • 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[#]&]

A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

Original entry on oeis.org

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, 223, 1, 240, 17, 247, 1, 305, 1, 286, 23
Offset: 0

Views

Author

Gus Wiseman, Jan 01 2019

Keywords

Comments

First differs from A072721 at a(12) = 9, A072721(12) = 10.
First differs from A379957 at a(16) = 10, A379957(16) = 9.

Examples

			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).

Links

Crossrefs

Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912, A379957.

Programs

  • 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}]
  • PARI
    a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&issquarefree(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025
  • PARI
    seq(n)={Vec(1 + sum(d=2, n, if(issquarefree(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025

Formula

From Andrew Howroyd, Jan 23 2025: (Start)
G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A005117(k)^j)).
a(p) = 1 for prime p. (End)

Extensions

a(66) onwards from Andrew Howroyd, Jan 23 2025
Showing 1-6 of 6 results.

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