A291445 -id:A291445 - cp's OEIS frontend

A291355 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, j divides Sum_{i=1..j} s_i^3.

1, 1, 0, 2, 4, 8, 0, 8, 16, 24, 0, 46, 46, 46, 0, 218, 1658, 6542, 0, 0, 2172, 6200, 0, 0, 0, 0, 0, 1652, 5878, 26778, 0, 6242, 6242, 6242, 0, 0, 0, 179878, 0, 169024, 472924, 603878, 0, 123100, 123100, 758560, 0, 0, 0, 0, 0, 0, 244698, 489396, 0, 495512

Offset: 0

Seiichi Manyama, Aug 23 2017

nonn

a(n) = 0 if n == 2 mod 4 as in this case n does not divide (n(n+1)/2)^2. In addition, a(4m+2) <= a(4m+3) <= a(4m+4) <= a(4m+5) for all m. - Chai Wah Wu, Aug 23 2017

The similar sequence b(n) in which the element of the permutation are squared instead of cubed, seems much more sparse. For 1 < n <= 250, the only nonzero terms are b(11)=12 and b(23)=480. - Giovanni Resta, Aug 23 2017

			1 divides 1^3,
2 divides 1^3 + 3^3,
3 divides 1^3 + 3^3 + 2^3,
4 divides 1^3 + 3^3 + 2^3 + 4^3.
So [1, 3, 2, 4] satisfies all the conditions.
---------------------------------------------
a(1) = 1: [[1]];
a(3) = 2: [[1, 3, 2], [3, 1, 2]];
a(4) = 4: [[1, 3, 2, 4], [2, 4, 3, 1], [3, 1, 2, 4], [4, 2, 3, 1]];
a(5) = 8: [[1, 3, 2, 4, 5], [2, 4, 3, 1, 5], [2, 4, 3, 5, 1], [3, 1, 2, 4, 5], [3, 5, 4, 2, 1], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [5, 3, 4, 2, 1]].

Giovanni Resta, Table of n, a(n) for n = 0..100

Cf. A291445.

def search(a, sum, k, size, num)
  if num == size + 1
    @cnt += 1
  else
    (1..size).each{|i|
      if a[i - 1] == 0 && (sum + i ** k) % num == 0
        a[i - 1] = 1
        search(a, sum + i ** k, k, size, num + 1)
        a[i - 1] = 0
      end
    }
  end
end
def A(k, n)
  a = [0] * n
  @cnt = 0
  search(a, 0, k, n, 1)
  @cnt
end
def A291355(n)
  (0..n).map{|i| A(3, i)}
end
p A291355(20)

a(0), a(13)-a(30) from Alois P. Heinz, Aug 23 2017

a(31)-a(55) from Giovanni Resta, Aug 23 2017

A291519 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_n = 1 (if n>0) and such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.

Original entry on oeis.org

1, 1, 1, 2, 6, 18, 42, 90, 228, 498, 1152, 2274, 5460, 10308, 20868, 39222, 78126, 151092, 306144, 596796, 1204734, 2359518, 4720854, 9229200, 18329442, 35889966, 71284524, 140430234, 279790956, 554351988, 1105988208, 2195249184, 4371548958, 8665192968

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

Appears to approximately double (for n > 1) for each successive n. - Chai Wah Wu, Aug 26 2017

			5                 divides 5^3,
5 + 4             divides 5^3 + 4^3,
5 + 4 + 3         divides 5^3 + 4^3 + 3^3,
5 + 4 + 3 + 2     divides 5^3 + 4^3 + 3^3 + 2^3,
5 + 4 + 3 + 2 + 1 divides 5^3 + 4^3 + 3^3 + 2^3 + 1^3.
So [5, 4, 3, 2, 1] satisfies all the conditions.
-------------------------------------------------------
a(1) = 1: [[1]];
a(2) = 1: [[2, 1]];
a(3) = 2: [[2, 3, 1], [3, 2, 1]];
a(4) = 6: [[2, 3, 4, 1], [2, 4, 3, 1], [3, 2, 4, 1], [3, 4, 2, 1], [4, 2, 3, 1], [4, 3, 2, 1]];
a(5) = 18: [[2, 3, 4, 5, 1], [2, 3, 5, 4, 1], [2, 4, 3, 5, 1], [2, 5, 3, 4, 1], [3, 2, 4, 5, 1], [3, 2, 5, 4, 1], [3, 4, 2, 5, 1], [3, 4, 5, 2, 1], [3, 5, 2, 4, 1], [3, 5, 4, 2, 1], [4, 2, 3, 5, 1], [4, 3, 2, 5, 1], [4, 3, 5, 2, 1], [4, 5, 3, 2, 1], [5, 2, 3, 4, 1], [5, 3, 2, 4, 1], [5, 3, 4, 2, 1], [5, 4, 3, 2, 1]].

Chai Wah Wu, Table of n, a(n) for n = 0..120

Cf. A291445, A291518.

A291445(n) >= a(n) + A291518(n) for n > 1.

a(0), a(14)-a(33) from Alois P. Heinz, Aug 25 2017

A291518 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_n = n (if n>0) and such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.

Original entry on oeis.org

1, 1, 1, 2, 6, 12, 30, 78, 186, 414, 912, 2064, 4338, 9798, 20106, 40974, 80196, 158322, 309414, 615558, 1212402, 2417136, 4776654, 9497508, 18726708, 37056150, 72946116, 144230640, 284660874, 564451830, 1118803818, 2224792026, 4420041210, 8791590168

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

			1                 divides 1^3,
1 + 2             divides 1^3 + 2^3,
1 + 2 + 3         divides 1^3 + 2^3 + 3^3,
1 + 2 + 3 + 4     divides 1^3 + 2^3 + 3^3 + 4^3,
1 + 2 + 3 + 4 + 5 divides 1^3 + 2^3 + 3^3 + 4^3 + 5^3.
So [1, 2, 3, 4, 5] satisfies all the conditions.
-------------------------------------------------------
a(1) = 1: [[1]];
a(2) = 1: [[1, 2]];
a(3) = 2: [[1, 2, 3], [2, 1, 3]];
a(4) = 6: [[1, 2, 3, 4], [1, 3, 2, 4], [2, 1, 3, 4], [2, 3, 1, 4], [3, 1, 2, 4], [3, 2, 1, 4]];
a(5) = 12: [[1, 2, 3, 4, 5], [1, 3, 2, 4, 5], [2, 1, 3, 4, 5], [2, 3, 1, 4, 5], [2, 3, 4, 1, 5], [2, 4, 3, 1, 5], [3, 1, 2, 4, 5], [3, 2, 1, 4, 5], [3, 2, 4, 1, 5], [3, 4, 2, 1, 5], [4, 2, 3, 1, 5], [4, 3, 2, 1, 5]].

Cf. A291445, A291519.

a(n+1) = A291445(n).

A291445(n) >= a(n) + A291519(n) for n > 1.

cp's OEIS Frontend

A291355 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, j divides Sum_{i=1..j} s_i^3.

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A291519 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_n = 1 (if n>0) and such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.

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A291518 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_n = n (if n>0) and such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.

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