A330590 Triangle read by rows: T(n,k) is the number of positive integers m dividing x^n - x^k for all integers x, 0 < k < n.

2, 4, 2, 2, 6, 2, 8, 2, 8, 2, 2, 12, 2, 8, 2, 8, 2, 16, 2, 8, 2, 2, 18, 2, 20, 2, 8, 2, 8, 2, 24, 2, 20, 2, 8, 2, 2, 12, 2, 24, 2, 20, 2, 8, 2, 8, 2, 16, 2, 24, 2, 20, 2, 8, 2, 2, 12, 2, 20, 2, 24, 2, 20, 2, 8, 2, 32, 2, 16, 2, 24, 2, 24, 2, 20, 2, 8, 2, 2, 72

Offset

2,1

Links

Peter Kagey, Table of n, a(n) for n = 2..10012 (first 141 rows, flattened)

Formula

T(n,k) = A000005(A330541(n,k)).

Conjecture: T(n,1) = 2^A067513(n-1).

Example

Table begins:
  n\k| 1   2   3   4   5   6   7   8   9  10  11
  ---+-------------------------------------------------
   2 | 2;
   3 | 4,  2;
   4 | 2,  6,  2;
   5 | 8,  2,  8,  2;
   6 | 2, 12,  2,  8,  2;
   7 | 8,  2, 16,  2,  8,  2;
   8 | 2, 18,  2, 20,  2,  8,  2;
   9 | 8,  2, 24,  2, 20,  2,  8,  2;
  10 | 2, 12,  2, 24,  2, 20,  2,  8,  2;
  11 | 8,  2, 16,  2, 24,  2, 20,  2,  8,  2;
  12 | 2, 12,  2, 20,  2, 24,  2, 20,  2,  8,  2.
For n=4 and k=2, the sequence x^4 - x^2 evaluated on the positive (equivalently, negative) integers is 0,12,72,240,600,1260,2352,4032,6480,9900,... and all terms are divisible by the following T(4,2) = 6 positive integers: 1, 2, 3, 4, 6, and 12.

Crossrefs

Cf. A000005, A330541.

Sequence in context: A368435 A058384 A255671 * A055097 A258751 A280233

Adjacent sequences: A330587 A330588 A330589 * A330591 A330592 A330593

Keyword

nonn,tabl

Author

Peter Kagey, Dec 18 2019

Status

approved