A073090 -id:A073090 - cp's OEIS frontend

A349145 Number of ordered n-tuples (x_1, x_2, x_3, ..., x_n) such that Sum_{k=1..n} k/x_k is an integer and x_k is an integer between 1 and n for 1 <= k <= n.

1, 1, 2, 8, 43, 207, 2391, 15539, 182078, 2070189, 35850460, 338695058, 10609401552, 115445915555

Offset: 0

Seiichi Manyama, Nov 08 2021

			1/1 + 2/1 = 3 and 3 is an integer.
1/1 + 2/2 = 2 and 2 is an integer.
1/2 + 2/1 = 5/2.
1/2 + 2/2 = 3/2.
So a(2) = 2.

Cf. A073090, A349146.

from fractions import Fraction
from itertools import product
def A349145(n): return sum(1 for d in product(range(1,n+1),repeat=n) if sum(Fraction(i+1,j) for i, j in enumerate(d)).denominator == 1) # Chai Wah Wu, Nov 09 2021

def A(n)
  return 1 if n == 0
  cnt = 0
  (1..n).to_a.repeated_permutation(n){|i|
    cnt += 1 if (1..n).inject(0){|s, j| s + j / i[j - 1].to_r}.denominator == 1
  }
  cnt
end
def A349145(n)
  (0..n).map{|i| A(i)}
end
p A349145(6)

a(10)-a(13) from Alois P. Heinz, Nov 08 2021

A349257 Largest integer that can be expressed as Sum_{k=1..n} k/p(k), where p is a permutation of [n].

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 11, 15, 18, 21, 22, 27, 28, 32, 36, 40, 41, 46, 47

Offset: 0

Seiichi Manyama, Nov 12 2021

Cf. A022819, A073090, A349277.

def A(n)
  max = 0
  (1..n).to_a.permutation{|i|
    m = (1..n).inject(0){|s, j| s + j / i[j - 1].to_r}
    if m.denominator == 1
      max = m if max < m
    end
  }
  max.to_i
end
def A349257(n)
  (0..n).map{|i| A(i)}
end
p A349257(8)

a(n) = 1 + a(n-1) if n is prime. - Alois P. Heinz, Nov 12 2021

a(12)-a(19) from Alois P. Heinz, Nov 12 2021

A349277 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of permutations p of [n] such that Sum_{j=1..n} j/p(j) is an integer and p(n) = k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 8, 4, 4, 2, 2, 0, 2, 0, 8, 18, 18, 14, 18, 0, 14, 0, 0, 22, 113, 130, 102, 135, 108, 122, 0, 314, 0, 104, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1128, 1152, 1166, 1130, 1078, 1334, 1182, 0, 1734, 3390, 1226, 0, 1128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14520

Offset: 1

Seiichi Manyama, Nov 12 2021

			Triangle begins:
    1;
    0,   1;
    0,   0,   1;
    1,   0,   0,   1;
    0,   0,   0,   0,   2;
    1,   1,   2,   2,   0,   2;
    0,   0,   0,   0,   0,   0, 8;
    4,   4,   2,   2,   0,   2, 0,   8;
   18,  18,  14,  18,   0,  14, 0,   0, 22;
  113, 130, 102, 135, 108, 122, 0, 314,  0, 104;
    0,   0,   0,   0,   0,   0, 0,   0,  0,   0, 1128;

Row sum gives A073090.

def A(n)
  ary = Array.new(n, 0)
  (1..n).to_a.permutation{|i|
    ary[i[-1] - 1] += 1 if (1..n).inject(0){|s, j| s + j / i[j - 1].to_r}.denominator == 1
  }
  ary
end
def A349277(n)
  (1..n).map{|i| A(i)}.flatten
end
p A349277(8)

If n is prime, T(n,k) = 0 for 1 <= k <= n-1.

T(n,n) = A073090(n-1).

cp's OEIS Frontend

A349145 Number of ordered n-tuples (x_1, x_2, x_3, ..., x_n) such that Sum_{k=1..n} k/x_k is an integer and x_k is an integer between 1 and n for 1 <= k <= n.

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A349257 Largest integer that can be expressed as Sum_{k=1..n} k/p(k), where p is a permutation of [n].

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A349277 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of permutations p of [n] such that Sum_{j=1..n} j/p(j) is an integer and p(n) = k.

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Examples

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Programs

Ruby

Formula