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.

A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 1, 1, 7, 1, 1, 7, 5, 1, 1, 1, 1, 11, 15, 12, 6, 1, 1, 1, 1, 15, 7, 1, 1, 30, 19, 1, 1, 22, 34, 8, 1, 1, 1, 1, 30, 58, 27, 9, 1, 1, 1, 1, 42, 84, 64, 10, 1, 1, 105, 37, 1, 1, 56, 11, 1
Offset: 1

Views

Author

Gus Wiseman, Sep 13 2019

Keywords

Examples

			Triangle begins:
  1
  1  1
  1  1
  1  2  1
  1  1
  1  3  3  1
  1  1
  1  5  4  1
  1  7  1
  1  7  5  1
  1  1
  1 11 15 12  6  1
  1  1
  1 15  7  1
  1 30 19  1
  1 22 34  8  1

Crossrefs

Row sums are A067538.
The version for subsets is A327481.
Cf. A027750, A066571, A237984, A240850, A327474, A327483, A327484.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Mean[#]==d&]],{n,20},{d,Divisors[n]}]

Extensions

Name edited by Peter Munn, Mar 05 2025

A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.

Original entry on oeis.org

1, 2, 3, 6, 9, 16, 27, 46, 77, 136, 239, 426, 769, 1400, 2571, 4762, 8857, 16568, 31139, 58734, 111165, 211044, 401695, 766418, 1465489, 2807672, 5388783, 10359850, 19946833, 38459624, 74251095, 143524762, 277742489, 538043664, 1043333935, 2025040766, 3933915349
Offset: 0

Views

Author

Gus Wiseman, Sep 13 2019

Keywords

Examples

			The a(0) = 1 through a(5) = 16 subsets:
  {}  {}   {}   {}       {}       {}
      {1}  {1}  {1}      {1}      {1}
           {2}  {2}      {2}      {2}
                {3}      {3}      {3}
                {1,3}    {4}      {4}
                {1,2,3}  {1,3}    {5}
                         {2,4}    {1,3}
                         {1,2,3}  {1,5}
                         {2,3,4}  {2,4}
                                  {3,5}
                                  {1,2,3}
                                  {1,3,5}
                                  {2,3,4}
                                  {3,4,5}
                                  {1,2,4,5}
                                  {1,2,3,4,5}

Crossrefs

If the subset is required to contain n, we get A063776.
Cf. A000016, A051293, A065795, A082550, A135342, A327474, A327481.

Programs

  • Maple
    with(numtheory):
b:= n-> add(2^(n/d)*phi(d), d=select(x-> x::odd, divisors(n)))/n:
a:= proc(n) option remember; `if`(n=0, 1, b(n)-1+a(n-1)) end:
seq(a(n), n=0..36);  # Alois P. Heinz, Jan 13 2024
  • Mathematica
    Table[Length[Select[Subsets[Range[n]],#=={}||IntegerQ[Mean[#]]&]],{n,0,10}]
  • Python
    from sympy import totient, divisors
def A327475(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))-n+1 # Chai Wah Wu, Feb 22 2023

Formula

a(n) = A051293(n) + 1.

A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 7, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 19, 9, 3, 1, 1, 3, 9, 25, 25, 9, 3, 1, 1, 3, 9, 29, 51, 29, 9, 3, 1, 1, 3, 9, 31, 75, 75, 31, 9, 3, 1, 1, 3, 9, 31, 93, 151, 93, 31, 9, 3, 1, 1, 3, 9, 31, 105, 235, 235, 105, 31, 9, 3, 1
Offset: 1

Views

Author

Gus Wiseman, Sep 13 2019

Keywords

Comments

All terms are odd.

Examples

			Triangle begins:
                         1
                       1   1
                     1   3   1
                   1   3   3   1
                 1   3   7   3   1
               1   3   9   9   3   1
             1   3   9  19   9   3   1
           1   3   9  25  25   9   3   1
         1   3   9  29  51  29   9   3   1
       1   3   9  31  75  75  31   9   3   1
     1   3   9  31  93 151  93  31   9   3   1
   1   3   9  31 105 235 235 105  31   9   3   1
The subsets counted in row n = 5:
  {1}  {2}      {3}          {4}      {5}
       {1,3}    {1,5}        {3,5}
       {1,2,3}  {2,4}        {3,4,5}
                {1,3,5}
                {2,3,4}
                {1,2,4,5}
                {1,2,3,4,5}

Crossrefs

Row sums are A051293.
The sequence of rows converges to A066571.
The version for partitions is A327482.
Cf. A000016, A063776, A065795, A135342, A327474, A327475, A327483, A327484.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Mean[#]==k&]],{n,10},{k,n}]

A327478 Numbers whose average binary index is also a binary index.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 21, 28, 31, 32, 39, 42, 56, 57, 62, 64, 73, 78, 84, 93, 107, 112, 114, 124, 127, 128, 141, 146, 155, 156, 168, 175, 177, 186, 214, 217, 224, 228, 245, 248, 254, 256, 267, 273, 282, 287, 292, 310, 312, 313, 336, 341, 350, 354, 371, 372
Offset: 1

Views

Author

Gus Wiseman, Sep 13 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Examples

			The sequence of terms together with their binary indices begins:
   1: 1
   2: 2
   4: 3
   7: 1 2 3
   8: 4
  14: 2 3 4
  16: 5
  21: 1 3 5
  28: 3 4 5
  31: 1 2 3 4 5
  32: 6
  39: 1 2 3 6
  42: 2 4 6
  56: 4 5 6
  57: 1 4 5 6
  61: 2 3 4 5 6

Crossrefs

Numbers whose binary indices have integer mean are A326669.
Cf. A000016, A000120, A029931, A048793, A065795, A070939, A237984, A240850, A327473, A327474, A327481.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],MemberQ[bpe[#],Mean[bpe[#]]]&]

A327484 Number of integer partitions of 2^n whose mean is a power of 2.

Original entry on oeis.org

1, 2, 4, 11, 66, 1417, 178803, 275379307, 15254411521973, 108800468645440803267, 964567296140908420613296779144, 219614169629364529542990295052656098001967511, 38626966436500261962963100479469496821891576834974275502742922521
Offset: 0

Views

Author

Gus Wiseman, Sep 13 2019

Keywords

Comments

Number of partitions of 2^n whose number of parts is a power of 2. - Chai Wah Wu, Sep 21 2023

Examples

			The a(0) = 1 through a(3) = 11 partitions:
  (1)  (2)   (4)     (8)
       (11)  (22)    (44)
             (31)    (53)
             (1111)  (62)
                     (71)
                     (2222)
                     (3221)
                     (3311)
                     (4211)
                     (5111)
                     (11111111)

Links

Crossrefs

Row sums of A327483.
Cf. A067538, A068413, A135342, A237984, A327474, A327481, A327482.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2^n],IntegerQ[Mean[#]]&]],{n,0,5}]
  • Python
    from sympy.utilities.iterables import partitions
def A327484(n): return sum(1 for s,p in partitions(1<Chai Wah Wu, Sep 21 2023
  • Python
    # uses A008284_T
def A327484(n): return sum(A008284_T(1<Chai Wah Wu, Sep 21 2023

Extensions

a(7) from Chai Wah Wu, Sep 14 2019
a(8)-a(11) from Alois P. Heinz, Sep 21 2023
a(12) from Chai Wah Wu, Sep 21 2023

A327485 Product of means of integer partitions with Heinz numbers from 2 to n.

Original entry on oeis.org

1, 2, 2, 6, 9, 36, 36, 72, 144, 720, 960, 5760, 14400, 36000, 36000, 252000, 420000, 3360000, 5600000, 16800000, 50400000, 453600000, 567000000, 1701000000, 5953500000, 11907000000, 23814000000, 238140000000, 476280000000, 5239080000000, 5239080000000
Offset: 2

Views

Author

Gus Wiseman, Sep 28 2019

Keywords

Comments

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.

Examples

			We have a(7) = 1 * 2 * 1 * 3 * 3/2 * 4 = 36.

Crossrefs

Partial products of A326567/A326568.
Cf. A056239, A112798, A316413, A327474, A327482, A327486.

Programs

  • Mathematica
    Table[Product[Total[Cases[FactorInteger[k],{p_,k_}:>k*PrimePi[p]]]/PrimeOmega[k],{k,2,n}],{n,2,30}]

Formula

a(n > 2) = a(n - 1) * A326567(n) / A326568(n).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.