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

2,4

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.

Links

Table of n, a(n) for n=2..62.

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

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.

Example

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

Crossrefs

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

Sequence in context: A318921 A114516 A218788 * A027185 A035641 A242434

Adjacent sequences: A328966 A328967 A328968 * A328970 A328971 A328972

Keyword

nonn,tabf

Author

Jonathan Blanchette, Nov 01 2019

Status

approved