A354969 For prime partition (k_1, k_2, ..., k_i) of the number x, let the conjugate partition be (m_1, m_2, ... , m_j). If n = k_1 * k_2 * ... * k_i, then a(n) = m1 * m2 * ... * m_j.

1, 1, 4, 1, 4, 1, 9, 8, 4, 1, 9, 1, 4, 8, 16, 1, 18, 1, 9, 8, 4, 1, 16, 32, 4, 27, 9, 1, 18, 1, 25, 8, 4, 32, 32, 1, 4, 8, 16, 1, 18, 1, 9, 27, 4, 1, 25, 128, 72, 8, 9, 1, 48, 32, 16, 8, 4, 1, 32, 1, 4, 27, 36, 32, 18, 1, 9, 8, 72, 1, 50, 1, 4, 108, 9, 128, 18, 1, 25, 64, 4, 1, 32, 32, 4, 8, 16, 1, 48, 128, 9, 8, 4, 32, 36, 1, 288, 27, 128

Offset

2,3

Links

Seiichi Manyama, Table of n, a(n) for n = 2..10000

Eric Weisstein's World of Mathematics, Prime Partition

Formula

If p is prime, a(p^e) = e^p.

Example

6 = 3*2.
x x x -> 3
x x   -> 2
2*2*1=4. So a(6) = 4.
8 = 2*2*2.
x x -> 2
x x -> 2
x x -> 2
3*3=9. So a(8) = 9.
9 = 3*3.
x x x -> 3
x x x -> 3
2*2*2=8. So a(9) = 8.
10 = 5*2
x x x x x -> 5
x x       -> 2
2*2*1*1*1=4. So a(10) = 4.

Prog

(Ruby)
require 'prime'
def A(ary)
  y = ary.size
  x = ary[0]
  a = (0..y - 1).map{|i| [1] * ary[i] + [0] * (x - ary[i])}
  (0..x - 1).map{|i| (0..y - 1).inject(0){|s, j| s + a[j][i]}}
end
def B(n)
  n.prime_division.map{|i| [i[0]] * i[1]}.flatten
end
def C(n)
  A(B(n).reverse).inject(:*)
end
def A354969(n)
  (2..n).map{|i| C(i)}
end
p A354969(100)

Crossrefs

Sequence in context: A322820 A097936 A277027 * A050338 A301598 A077088

Adjacent sequences: A354966 A354967 A354968 * A354970 A354971 A354972

Keyword

nonn

Author

Seiichi Manyama, Jun 14 2022

Status

approved