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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, 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
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 14 2022
STATUS
approved