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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A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

References

  • G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Links

Crossrefs

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Programs

  • Mathematica
    Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)
  • PARI
    A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

Formula

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020
As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786
Offset: 1

Views

Author

Gus Wiseman, May 07 2021

Keywords

Examples

			The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Crossrefs

Antidiagonal sums of the array (or row sums of the triangle) A334997.
A000005 counts divisors of n.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
Array version of A334997 has:
- column k = 2 A007425,
- transpose A077592,
- subdiagonal n = k + 1 A163767,
- strict case A343662 (row sums: A337256),
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Programs

  • Mathematica
    Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]

A343936 Number of ways to choose a multiset of n divisors of n - 1.

Original entry on oeis.org

1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884
Offset: 1

Views

Author

Gus Wiseman, May 05 2021

Keywords

Examples

			The a(1) = 1 through a(5) = 5 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
      {2}  {1,3}  {1,1,2}  {1,1,1,5}
           {3,3}  {1,1,4}  {1,1,5,5}
                  {1,2,2}  {1,5,5,5}
                  {1,2,4}  {5,5,5,5}
                  {1,4,4}
                  {2,2,2}
                  {2,2,4}
                  {2,4,4}
                  {4,4,4}
The a(6) = 56 multisets:
  11111  11136  11333  12236  13366  22266  23666
  11112  11166  11336  12266  13666  22333  26666
  11113  11222  11366  12333  16666  22336  33333
  11116  11223  11666  12336  22222  22366  33336
  11122  11226  12222  12366  22223  22666  33366
  11123  11233  12223  12666  22226  23333  33666
  11126  11236  12226  13333  22233  23336  36666
  11133  11266  12233  13336  22236  23366  66666

Crossrefs

The version for chains of divisors is A163767.
Diagonal n = k + 1 of A343658.
Choosing n divisors of n gives A343935.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343656, A343661.

Programs

  • Mathematica
    multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n-1],{n,50}]

Formula

a(n) = ((sigma(n - 1), n)) = binomial(sigma(n - 1) + n - 1, n) where sigma = A000005 and binomial = A007318.
Previous Showing 11-13 of 13 results.

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

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