%I #42 Jan 08 2020 00:14:14

%S 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,

%T 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

%N 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.

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

%C The sum of the rows equals the sequence A006022.

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

%H Jonathan Blanchette and Robert Laganière, <a href="https://arxiv.org/abs/1910.11749">A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting</a>, arXiv:1910.11749 [cs.DS], 2019.

%F 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:

%F 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;

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

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

%e First few rows are:

%e 1;

%e 0, 1;

%e 3, 0;

%e 0, 0, 1;

%e 3, 1, 0;

%e 0, 0, 0, 1;

%e 7, 0, 0, 0;

%e 0, 4, 0, 0;

%e 5, 0, 1, 0;

%e 0, 0, 0, 0, 1;

%e ...

%e Examples (see the p_k formulas)

%e T(2^3,1) = (2^3-1) / (2-1) = 7

%e T(3^2,1) = (3^2-1) / (3-1) = 4

%e T(3*2,2) = (6/(2*3)) * (3^2-1) / (3-1) = 4

%e T(12,1) = (12/(2^2)) * (2^2-1) / (2-1) = 9

%e T(12,2) = (12/(2^2*3)) * (3-1) / (3-1) = 1

%e T(15,2) = (15/3) * (3-1) / (3-1) = 5

%e T(15,3) = (15/(2^2*3)) * (3-1) / (3-1) = 1

%e T(2*3*5^2*7,3) = (2*3*5^2*7/(2*3*5^2)) * (5^2-1) / (5-1) = 42

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

%K nonn,tabf

%O 2,4

%A _Jonathan Blanchette_, Nov 01 2019