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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A329436 Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 5, 6, 8, 7, 13, 10, 16, 18, 22, 21, 34, 29, 44, 45, 56, 56, 82, 78, 100, 109, 136, 137, 185, 181, 231, 247, 295, 317, 399, 404, 490, 533, 638, 669, 817, 853, 1020, 1108, 1276, 1371, 1638, 1728, 2017, 2186, 2519, 2702, 3153, 3371, 3885
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A025147.
Number of uniform (constant multiplicity) partitions of n not containing 1, ranked by the odd terms of A072774. - Gus Wiseman, Dec 01 2023

Examples

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(2) = 1 through a(10) = 8 uniform partitions not containing 1:
  (2)  (3)  (4)    (5)    (6)      (7)    (8)        (9)      (10)
            (2,2)  (3,2)  (3,3)    (4,3)  (4,4)      (5,4)    (5,5)
                          (4,2)    (5,2)  (5,3)      (6,3)    (6,4)
                          (2,2,2)         (6,2)      (7,2)    (7,3)
                                          (2,2,2,2)  (3,3,3)  (8,2)
                                                     (4,3,2)  (5,3,2)
                                                              (3,3,2,2)
                                                              (2,2,2,2,2)
(End)

Crossrefs

The strict case is A025147.
The version allowing 1 is A047966.
The version requiring 1 is A097986.
Cf. A023645, A047968, A072774, A096765, A329435, A329436, A367586.

Programs

  • Mathematica
    nmax = 56; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&SameQ@@Length/@Split[#]&]], {n,0,30}] (* Gus Wiseman, Dec 01 2023 *)

Formula

G.f.: Sum_{k>=1} A025147(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A025147(d).

A343005 a(n) is the number of dihedral symmetries D_{2m} (m >= 3) that configurations of n non-overlapping equal circles can possess.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 3, 5, 5, 3, 5, 6, 4, 5, 5, 5, 7, 5, 3, 7, 8, 4, 5, 7, 5, 7, 7, 5, 7, 5, 5, 10, 8, 3, 5, 9, 7, 7, 7, 5, 9, 7, 3, 9, 10, 6, 7, 7, 5, 7, 9, 9, 9, 5, 3, 11, 11, 3, 7, 10, 8, 9, 7, 5, 7, 9, 7, 11, 11, 3, 7, 9, 7, 9, 7, 9, 12, 6, 3, 11, 13
Offset: 2

Views

Author

Ya-Ping Lu, Apr 02 2021

Keywords

Examples

			a(2) = 0 because the configuration of 2 circles only possesses D_{4} symmetry.
a(6) = 3 because configurations of 6 circles can have three dihedral symmetries: D_{12} (6 circles arranged in regular hexagon shape), D_{10} (5 circles arranged in regular pentagon shape and the other circle in the center of the pentagon), and D_{6} (6 circles arranged in equilateral triangle shape).

Crossrefs

Cf. A000005, A023645, A274010.

Programs

  • Mathematica
    Table[DivisorSigma[0,n]+DivisorSigma[0,n-1]-3,{n,2,85}] (* Stefano Spezia, Apr 06 2021 *)
  • Python
    from sympy import divisor_count
for n in range(2, 101):
    print(divisor_count(n) + divisor_count(n - 1) - 3, end=", ")

Formula

For n >= 2, a(n) = A274010(n) - 1 = A023645(n) + A023645(n-1) = tau(n) + tau(n-1) - 3, where tau(n) = A000005(n), the number of divisors of n.

A062558 Number of nonisomorphic cyclic subgroups of dihedral group with 2n elements.

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 3, 4, 4, 4, 3, 6, 3, 4, 5, 5, 3, 6, 3, 6, 5, 4, 3, 8, 4, 4, 5, 6, 3, 8, 3, 6, 5, 4, 5, 9, 3, 4, 5, 8, 3, 8, 3, 6, 7, 4, 3, 10, 4, 6, 5, 6, 3, 8, 5, 8, 5, 4, 3, 12, 3, 4, 7, 7, 5, 8, 3, 6, 5, 8, 3, 12, 3, 4, 7, 6, 5, 8, 3, 10, 6, 4, 3, 12, 5, 4, 5, 8, 3, 12, 5, 6, 5, 4, 5, 12, 3, 6, 7, 9, 3
Offset: 1

Views

Author

Vladeta Jovovic, Jul 03 2001

Keywords

Links

Crossrefs

Cf. A000005, A000045, A023645, A062249 (labeled case).
One more than A076984.

Programs

Formula

a(n) = A000005(n) + A000035(n) = tau(n)+(n mod 2), where tau(n) = the number of divisors of n.

A296966 Sum of all the parts in the partitions of n into two distinct parts such that the smaller part divides the larger.

Original entry on oeis.org

0, 0, 3, 4, 5, 12, 7, 16, 18, 20, 11, 48, 13, 28, 45, 48, 17, 72, 19, 80, 63, 44, 23, 144, 50, 52, 81, 112, 29, 180, 31, 128, 99, 68, 105, 252, 37, 76, 117, 240, 41, 252, 43, 176, 225, 92, 47, 384, 98, 200, 153, 208, 53, 324, 165, 336, 171, 116, 59, 600, 61
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 22 2017

Keywords

Examples

			From _Wesley Ivan Hurt_, Feb 21 2018: (Start)
a(5) = 5; there is one partition of 5 into two distinct parts such that the smaller part divides the larger, namely (4,1), so the sum of the parts is then 4 + 1 = 5.
a(6) = 12; the partitions of 6 into two distinct parts such that the smaller part divides the larger are (5,1) and (4,2), and the sum of the parts is then 5 + 1 + 4 + 2 = 12.
a(7) = 7; there is one partition of 7 into two distinct parts such that the smaller part divides the larger, namely (6,1), so the sum of the parts is 6 + 1 = 7.
a(8) = 16; there are two partitions of 8 into 2 distinct parts such that the smaller divides the larger, namely (7,1) and (6,2). The sum of the parts is then 7 + 1 + 6 + 2 = 16.
(End)

Links

Crossrefs

Cf. A023645.

Programs

  • Mathematica
    Table[n*Sum[(Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
f[n_] := n*Length[Select[Divisors@n, 2 # < n &]]; Array[f, 61] (* or *)
f[n_] := Block[{t = DivisorSigma[0, n]}, n*If[OddQ@ n, t -1, t -2]]; Array[f, 61] (* Robert G. Wilson v, Dec 24 2017 *)
  • PARI
    a(n) = n*sum(i=1, floor((n-1)/2), floor(n/i) - floor((n-1)/i)) \\ Iain Fox, Dec 22 2017

Formula

a(n) = n * Sum_{i=1..floor((n-1)/2)} floor(n/i) - floor((n-1)/i).
a(n) = n * A023645(n). - Robert G. Wilson v, Dec 24 2017

A303972 Total volume of all cubes with side length n which can be split such that n = p + q, p divides q and p < q.

Original entry on oeis.org

0, 0, 27, 64, 125, 432, 343, 1024, 1458, 2000, 1331, 6912, 2197, 5488, 10125, 12288, 4913, 23328, 6859, 32000, 27783, 21296, 12167, 82944, 31250, 35152, 59049, 87808, 24389, 162000, 29791, 131072, 107811, 78608, 128625, 326592, 50653, 109744, 177957, 384000
Offset: 1

Views

Author

Wesley Ivan Hurt, May 03 2018

Keywords

Links

Crossrefs

Cf. A303873, A023645 (number of contributing cubes).

Programs

  • Magma
    [0, 0] cat [&+[(((n-k) div k)-(n-k-1) div k)*n^3: k in [1..(n-1) div 2]]: n in [3..80]]; // Vincenzo Librandi, May 04 2018
  • Maple
    A303972 := proc(n)
    v := 0 ;
    for p from 1 to n/2 do
        q := n-p ;
        if p < q and modp(q,p) = 0 then
            v := v+n^3 ;
        end if;
    end do:
    v ;
end proc:
seq(A303972(n),n=1..40) ; # R. J. Mathar, Jun 25 2018
  • Mathematica
    Table[n^3*Sum[(Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 50}]

Formula

a(n) = n^3 * Sum_{i=1..floor((n-1)/2)} floor((n-i)/i) - floor((n-i-1)/i).
a(n) = n * A303873(n).

A303973 Total volume of all rectangular prisms with dimensions (p,p,q) such that n = p + q, p divides q and p < q.

Original entry on oeis.org

0, 0, 2, 3, 4, 21, 6, 31, 62, 41, 10, 260, 12, 61, 372, 263, 16, 648, 18, 722, 868, 101, 22, 2292, 524, 121, 1700, 1544, 28, 3873, 30, 2135, 2964, 161, 2156, 7703, 36, 181, 4756, 6690, 40, 9051, 42, 4844, 11088, 221, 46, 18788, 2106, 5366, 10308, 7610, 52
Offset: 1

Views

Author

Wesley Ivan Hurt, May 03 2018

Keywords

Examples

			For n =12 the  prism (p,p,q) = (1,1,11) contributes 1*1*11=11 to the volume, (2,2,10) contributes 2*2*10= 40, (3,3,9) contributes 3*3*9= 81, (4,4,8) contributes 128. The total is a(12) = 11+40+81+128 = 260.

Links

Crossrefs

Cf. A303873, A023645 (number of contributing prisms).

Programs

  • Magma
    [0,0] cat [&+[k^2*(n-k)*(((n-k) div k)-((n-k-1) div k)):  k in [1..((n-1) div 2)]]: n in [3..80]]; // Vincenzo Librandi, May 04 2018
  • Maple
    A303973 := proc(n)
        v := 0 ;
        for p from 1 to n/2 do
                q := n-p ;
                if p < q and modp(q,p) = 0 then
                        v := v+p^2*q ;
                end if;
        end do:
        v ;
end proc:
seq(A303973(n),n=1..40) ; # R. J. Mathar, Jun 25 2018
  • Mathematica
    Table[Sum[i^2 (n - i) (Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^2 * (n-i) * (floor((n-i)/i) - floor((n-i-1)/i)).

A350801 a(n) = n*(tau(n) + 1) - 2*sigma(n) for n>=1, with a(0)=0.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 6, 5, 10, 10, 14, 9, 28, 11, 22, 27, 34, 15, 48, 17, 56, 41, 38, 21, 96, 38, 46, 55, 84, 27, 126, 29, 98, 69, 62, 79, 178, 35, 70, 83, 180, 39, 186, 41, 140, 159, 86, 45, 280, 82, 164, 111, 168, 51, 246, 131, 264, 125, 110, 57, 444, 59, 118, 233, 258, 157
Offset: 0

Views

Author

Wesley Ivan Hurt, Jan 16 2022

Keywords

Comments

Sum of the positive differences of the parts in the partitions of n into two parts such that the smaller part divides the larger (see example).

Examples

			a(10) = 14; The partitions of 10 into two parts such that the smaller divides the larger are (1,9), (2,8), and (5,5). The sum of the positive differences of the parts is then (9-1) + (8-2) + (5-5) = 14.

Links

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A023645, A032741.

Programs

  • Mathematica
    Join[{0}, Table[n (1 + DivisorSigma[0, n]) - 2*DivisorSigma[1, n], {n, 100}]]

Formula

For n > 0, a(n) = Sum_{d|n, d
For n > 0, a(n) = n*(A000005(n) + 1) - 2*A000203(n).
Previous Showing 21-27 of 27 results.

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