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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A343939 Number of n-chains of divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 49, 8, 165, 55, 121, 12, 1183, 14, 225, 256, 4845, 18, 3610, 20, 4851, 484, 529, 24, 73125, 351, 729, 4060, 12615, 30, 29791, 32, 435897, 1156, 1225, 1296, 494209, 38, 1521, 1600, 505981, 42, 79507, 44, 46575, 49726, 2209, 48
Offset: 1

Views

Author

Gus Wiseman, May 05 2021

Keywords

Examples

			The a(1) = 1 through a(5) = 6 chains:
  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)
       (2/1)  (3/1/1)  (2/1/1/1)  (5/1/1/1/1)
       (2/2)  (3/3/1)  (2/2/1/1)  (5/5/1/1/1)
              (3/3/3)  (2/2/2/1)  (5/5/5/1/1)
                       (2/2/2/2)  (5/5/5/5/1)
                       (4/1/1/1)  (5/5/5/5/5)
                       (4/2/1/1)
                       (4/2/2/1)
                       (4/2/2/2)
                       (4/4/1/1)
                       (4/4/2/1)
                       (4/4/2/2)
                       (4/4/4/1)
                       (4/4/4/2)
                       (4/4/4/4)

Crossrefs

Diagonal n = k - 1 of the array A077592.
Chains of length n - 1 are counted by A163767.
Diagonal n = k of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005(n) counts divisors of n.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k-1) counts strict k-chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict k-chains of divisors from n to 1.
A337255(n,k) counts strict k-chains of divisors starting with n.
A343658(n,k) counts k-multisets of divisors of n.
A343662(n,k) counts strict k-chains of divisors of n (row sums: A337256).
Cf. A062319, A066959, A143773, A176029, A327527, A343656.

Programs

  • Mathematica
    Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]

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}]

A186170 Number of prime factors times n minus sum of divisors.

Original entry on oeis.org

-1, -1, -1, 1, -1, 0, -1, 9, 5, 2, -1, 8, -1, 4, 6, 33, -1, 15, -1, 18, 10, 8, -1, 36, 19, 10, 41, 28, -1, 18, -1, 97, 18, 14, 22, 53, -1, 16, 22, 70, -1, 30, -1, 48, 57, 20, -1, 116, 41, 57, 30, 58, -1, 96, 38, 104, 34, 26, -1, 72, -1, 28, 85, 257, 46, 54, -1
Offset: 1

Views

Author

Irina Gerasimova, Jun 14 2013

Keywords

Comments

Places n where a(n) is prime are: 9, 10, 25, 27, 32, 36, 49, 64,...
Places n where a(n^2) is prime are: 3, 5, 6, 7, 8,...
Number of prime factors counted with multiplicity. - Harvey P. Dale, Jan 23 2019

Examples

			a(8) = (number of prime factors)*8 - (sum of divisors) = 3*8 - (1+2+4+8) = 24 - 15 = 9.

Links

Crossrefs

Cf. A000203, A001222, A066959.

Programs

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