A328969 Irregular table T(n,k), n >= 2, k=1..pi(n). arising in expressing the sequence A006022 as the coefficients depending on the maximal k-th prime factor pk of the formula for A006022(n) of its unique prime factor equation.

1, 0, 1, 3, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 7, 0, 0, 0, 0, 4, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 0, 0, 1, 0, 0, 0, 5, 1, 0, 0, 0, 15, 0, 0, 0, 0, 0

Offset: 2

Author

Jonathan Blanchette, Nov 01 2019

Keywords

• nonn
• tabf

Comments

The length of the n-th row is pi(n) (A000720), i.e., 1,2,2,3,... for n>2.

The sum of the rows equals the sequence A006022.

When n is prime the entire row is 0 except at p=n where T(p,p)=1.

Examples

			First few rows are:
  1;
  0, 1;
  3, 0;
  0, 0, 1;
  3, 1, 0;
  0, 0, 0, 1;
  7, 0, 0, 0;
  0, 4, 0, 0;
  5, 0, 1, 0;
  0, 0, 0, 0, 1;
  ...
Examples (see the p_k formulas)
T(2^3,1) = (2^3-1) / (2-1) = 7
T(3^2,1) = (3^2-1) / (3-1) = 4
T(3*2,2) = (6/(2*3)) * (3^2-1) / (3-1) = 4
T(12,1) = (12/(2^2)) * (2^2-1) / (2-1) = 9
T(12,2) = (12/(2^2*3)) * (3-1) / (3-1) = 1
T(15,2) = (15/3) * (3-1) / (3-1) = 5
T(15,3) = (15/(2^2*3)) * (3-1) / (3-1) = 1
T(2*3*5^2*7,3) = (2*3*5^2*7/(2*3*5^2)) * (5^2-1) / (5-1) = 42
		

Links

• Jonathan Blanchette and Robert Laganière, A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting, arXiv:1910.11749 [cs.DS], 2019.

Crossrefs

The rows sum to A006022. Cf. A129527 (first column).

Formula

Let p_k be the k-th prime, where k is the column index, p_k <= n, and n >= 2, and m_k is the multiplicity of p_k occurring in n:

T(n,p_k) = n * 1/(p_1^m_1*p_2^m_2*...*p_k^m_k) * (p_k^m_k-1)/(p_k-1), if p_k divides n;

T(n,p_k) = 0; if p_k does not divide n.

T(2*n,2) = A129527(n); T(2*n+1,2) = 0.