A291355 - OEIS

%I #45 Aug 26 2017 13:19:43

%S 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,

%T 0,0,1652,5878,26778,0,6242,6242,6242,0,0,0,179878,0,169024,472924,

%U 603878,0,123100,123100,758560,0,0,0,0,0,0,244698,489396,0,495512

%N 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.

%C 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

%C 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

%H Giovanni Resta, <a href="/A291355/b291355.txt">Table of n, a(n) for n = 0..100</a>

%e 1 divides 1^3,

%e 2 divides 1^3 + 3^3,

%e 3 divides 1^3 + 3^3 + 2^3,

%e 4 divides 1^3 + 3^3 + 2^3 + 4^3.

%e So [1, 3, 2, 4] satisfies all the conditions.

%e ---------------------------------------------

%e a(1) = 1: [[1]];

%e a(3) = 2: [[1, 3, 2], [3, 1, 2]];

%e a(4) = 4: [[1, 3, 2, 4], [2, 4, 3, 1], [3, 1, 2, 4], [4, 2, 3, 1]];

%e 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]].

%o (Ruby)

%o def search(a, sum, k, size, num)

%o   if num == size + 1

%o     @cnt += 1

%o   else

%o     (1..size).each{|i|

%o       if a[i - 1] == 0 && (sum + i ** k) % num == 0

%o         a[i - 1] = 1

%o         search(a, sum + i ** k, k, size, num + 1)

%o         a[i - 1] = 0

%o       end

%o     }

%o   end

%o end

%o def A(k, n)

%o   a = [0] * n

%o   @cnt = 0

%o   search(a, 0, k, n, 1)

%o   @cnt

%o end

%o def A291355(n)

%o   (0..n).map{|i| A(3, i)}

%o end

%o p A291355(20)

%Y Cf. A291445.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Aug 23 2017

%E a(0), a(13)-a(30) from _Alois P. Heinz_, Aug 23 2017

%E a(31)-a(55) from _Giovanni Resta_, Aug 23 2017