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User: Jonathan Blanchette

Jonathan Blanchette has authored 2 sequences.

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

Jonathan Blanchette, Nov 01 2019

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.

			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

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

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

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.

A328898 Sum of p-ary comparisons units required to rank a sequence in parallel when the sequence is partitioned into heaps equal to the prime factors p of the initial sequence length n.

Original entry on oeis.org

0, 1, 1, 6, 1, 11, 1, 28, 12, 27, 1, 58, 1, 51, 28, 120, 1, 105, 1, 154, 52, 123, 1, 260, 30, 171, 117, 298, 1, 281, 1, 496, 124, 291, 54, 534, 1, 363, 172, 708, 1, 545, 1, 730, 309, 531, 1, 1096, 56, 685, 292, 1018, 1, 963, 126, 1380, 364, 843, 1, 1462, 1, 963, 597, 2016, 174, 1337, 1, 1738, 532, 1333, 1, 2364, 1, 1371, 715, 2170, 128, 1865

Offset: 1

Jonathan Blanchette, Oct 30 2019

nonn

a(1) = 0.
1 <= a(n) <= (n^2-n)/2 for all n>1.
a(n) = (n^2-n)/2, if n is a power of 2.
a(n) = 1, if n is a prime number.

Antti Karttunen, Table of n, a(n) for n = 1..32768
Jonathan Blanchette and Robert Laganière, A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting, arXiv:1910.11749 [cs.DS], 2019.

Cf. A006022.

a(n) = my(f=factor(n)); n^2*sum(i=1, #f~, (1/f[i,1]) * (1/(prod(j=1, i, f[j,1]^f[j,2]))) * (f[i,1]^f[i,2]-1)/(f[i,1]-1)); \\ Michel Marcus, Oct 31 2019

a(n) = n^2*Sum_{i=1..m} 1/(f(i)^2*f(i-1)*f(i-2)*...*f(1)) where f(i) is the i-th prime factor of n with repetition and m is the number of prime factors.

a(n) = n^2*Sum_{p(i)}(1/p(i) * 1/(Product_{j=1..i} p(j)^k(j)) * (p(i)^k(i)-1)/(p(i)-1)) where p(i) is the i-th unique prime factor of n with multiplicity k(i) and p(i)

More terms from Antti Karttunen, Apr 12 2020

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User: Jonathan Blanchette

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.

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