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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 14 07:04 EDT 2020. Contains 336477 sequences. (Running on oeis4.)