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.

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A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 5, 6, 8, 8, 12, 12, 14, 16, 19, 19, 23, 23, 27, 29, 31, 31, 37, 38, 40, 42, 46, 46, 52, 52, 56, 58, 60, 62, 69, 69, 71, 73, 79, 79, 85, 85, 89, 93, 95, 95, 103, 104, 108, 110, 114, 114, 120, 122, 128, 130, 132, 132, 142
Offset: 0

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Author

Gus Wiseman, May 04 2019

Keywords

Comments

Also the number of integer partitions of n that are not hooks but whose augmented differences are hooks (original name). The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and otherwise aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
This sequence counts integer partitions with any number of ones and one part > 1 which appears at least twice. The Heinz numbers of these partitions are given by A325359.

Examples

			The a(4) = 1 through a(10) = 8 partitions:
  (22)  (221)  (33)    (331)    (44)      (333)      (55)
               (222)   (2221)   (2222)    (441)      (3331)
               (2211)  (22111)  (3311)    (22221)    (4411)
                                (22211)   (33111)    (22222)
                                (221111)  (222111)   (222211)
                                          (2211111)  (331111)
                                                     (2221111)
                                                     (22111111)

Links

Crossrefs

Cf. A049988, A093641, A325349, A325351, A325355, A325356, A325357, A325358.
Cf. A070824, A006218.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 0,
      numtheory[tau](n)-2+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 11 2019
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,y__,1...}/;x>1&&SameQ[x,y]]&]],{n,0,30}]
(* Second program: *)
a[n_] := a[n] = If[n<2, 0, DivisorSigma[0, n] - 2 + a[n-1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
  • Python
    from math import isqrt
def A325459(n): return 0 if n == 0 else (lambda m: 2*(sum(n//k for k in range(1, m+1))-n)+(1-m)*(1+m))(isqrt(n)) # Chai Wah Wu, Oct 07 2021

Formula

From M. F. Hasler, Oct 11 2019: (Start)
a(n) = A006218(n) - 2*n + 1, in terms of partial sums of number of divisors.
a(n) = Sum_{k=1..n} A070824(k): partial sums of A070824 = number of nontrivial divisors. (End)

Extensions

Name changed at the suggestion of Patrick James Smalley-Wall and Luc Rousseau by Gus Wiseman, Oct 11 2019

A329143 Number of integer partitions of n whose augmented differences are a periodic word.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 2, 2, 4, 4, 5, 3, 5, 2, 10, 5, 6, 5, 10, 5, 11, 7, 13, 6, 15, 6, 20, 11, 18, 12, 27, 8, 27, 16, 32, 14, 35, 14, 42, 23, 43, 17, 56, 17, 61, 31, 67, 25, 78, 28, 88, 41, 89, 35, 119, 39, 116, 60, 131, 52, 154, 52, 170, 75, 182
Offset: 0

Views

Author

Gus Wiseman, Nov 10 2019

Keywords

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
A finite sequence is periodic if its cyclic rotations are not all different.

Examples

			The a(n) partitions for n = 2, 5, 8, 14, 16, 22:
  11  32     53        95              5533              7744
      11111  3221      5432            7441              9652
             11111111  32222111        533311            554332
                       11111111111111  33222211          54333211
                                       1111111111111111  332222221111
                                                         1111111111111111111111

Crossrefs

The Heinz numbers of these partitions are given by A329132.
The aperiodic version is A329136.
The non-augmented version is A329144.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Cf. A000740, A027375, A328594, A329133, A329135, A325351, A325356, A329134.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

Formula

a(n) + A329136(n) = A000041(n).

Extensions

More terms from Jinyuan Wang, Jun 27 2020

A329137 Number of integer partitions of n whose differences are an aperiodic word.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 14, 20, 25, 39, 54, 69, 99, 130, 167, 224, 292, 373, 483, 620, 773, 993, 1246, 1554, 1946, 2421, 2987, 3700, 4548, 5575, 6821, 8330, 10101, 12287, 14852, 17935, 21599, 25986, 31132, 37295, 44539, 53112, 63212, 75123, 89055, 105503, 124682
Offset: 0

Views

Author

Gus Wiseman, Nov 09 2019

Keywords

Comments

A sequence is aperiodic if its cyclic rotations are all different.

Examples

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)    (3)    (4)      (5)        (6)          (7)
       (1,1)  (2,1)  (2,2)    (3,2)      (3,3)        (4,3)
                     (3,1)    (4,1)      (4,2)        (5,2)
                     (2,1,1)  (2,2,1)    (5,1)        (6,1)
                              (3,1,1)    (4,1,1)      (3,2,2)
                              (2,1,1,1)  (2,2,1,1)    (3,3,1)
                                         (3,1,1,1)    (4,2,1)
                                         (2,1,1,1,1)  (5,1,1)
                                                      (2,2,2,1)
                                                      (3,2,1,1)
                                                      (4,1,1,1)
                                                      (2,2,1,1,1)
                                                      (3,1,1,1,1)
                                                      (2,1,1,1,1,1)
With differences:
  ()  ()   ()   ()     ()       ()         ()
      (0)  (1)  (0)    (1)      (0)        (1)
                (2)    (3)      (2)        (3)
                (1,0)  (0,1)    (4)        (5)
                       (2,0)    (3,0)      (0,2)
                       (1,0,0)  (0,1,0)    (1,0)
                                (2,0,0)    (2,1)
                                (1,0,0,0)  (4,0)
                                           (0,0,1)
                                           (1,1,0)
                                           (3,0,0)
                                           (0,1,0,0)
                                           (2,0,0,0)
                                           (1,0,0,0,0)

Crossrefs

The Heinz numbers of these partitions are given by A329135.
The periodic version is A329144.
The augmented version is A329136.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is aperiodic are A329139.
Cf. A152061, A325356, A329132, A329133, A329134, A329140.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Table[Length[Select[IntegerPartitions[n],aperQ[Differences[#]]&]],{n,0,30}]

Formula

a(n) + A329144(n) = A000041(n).
Previous Showing 11-13 of 13 results.

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