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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.

%I A328969 #42 Jan 08 2020 00:14:14
%S A328969 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 A328969 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 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.
%C A328969 The length of the n-th row is pi(n) (A000720), i.e., 1,2,2,3,... for n>2.
%C A328969 The sum of the rows equals the sequence A006022.
%C A328969 When n is prime the entire row is 0 except at p=n where T(p,p)=1.
%H A328969 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 A328969 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 A328969 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 A328969 T(n,p_k) = 0; if p_k does not divide n.
%F A328969 T(2*n,2) = A129527(n); T(2*n+1,2) = 0.
%e A328969 First few rows are:
%e A328969   1;
%e A328969   0, 1;
%e A328969   3, 0;
%e A328969   0, 0, 1;
%e A328969   3, 1, 0;
%e A328969   0, 0, 0, 1;
%e A328969   7, 0, 0, 0;
%e A328969   0, 4, 0, 0;
%e A328969   5, 0, 1, 0;
%e A328969   0, 0, 0, 0, 1;
%e A328969   ...
%e A328969 Examples (see the p_k formulas)
%e A328969 T(2^3,1) = (2^3-1) / (2-1) = 7
%e A328969 T(3^2,1) = (3^2-1) / (3-1) = 4
%e A328969 T(3*2,2) = (6/(2*3)) * (3^2-1) / (3-1) = 4
%e A328969 T(12,1) = (12/(2^2)) * (2^2-1) / (2-1) = 9
%e A328969 T(12,2) = (12/(2^2*3)) * (3-1) / (3-1) = 1
%e A328969 T(15,2) = (15/3) * (3-1) / (3-1) = 5
%e A328969 T(15,3) = (15/(2^2*3)) * (3-1) / (3-1) = 1
%e A328969 T(2*3*5^2*7,3) = (2*3*5^2*7/(2*3*5^2)) * (5^2-1) / (5-1) = 42
%Y A328969 The rows sum to A006022. Cf. A129527 (first column).
%K A328969 nonn,tabf
%O A328969 2,4
%A A328969 _Jonathan Blanchette_, Nov 01 2019