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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.

%I A291355 #45 Aug 26 2017 13:19:43
%S A291355 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 A291355 0,0,1652,5878,26778,0,6242,6242,6242,0,0,0,179878,0,169024,472924,
%U A291355 603878,0,123100,123100,758560,0,0,0,0,0,0,244698,489396,0,495512
%N 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.
%C A291355 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 A291355 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 A291355 Giovanni Resta, <a href="/A291355/b291355.txt">Table of n, a(n) for n = 0..100</a>
%e A291355 1 divides 1^3,
%e A291355 2 divides 1^3 + 3^3,
%e A291355 3 divides 1^3 + 3^3 + 2^3,
%e A291355 4 divides 1^3 + 3^3 + 2^3 + 4^3.
%e A291355 So [1, 3, 2, 4] satisfies all the conditions.
%e A291355 ---------------------------------------------
%e A291355 a(1) = 1: [[1]];
%e A291355 a(3) = 2: [[1, 3, 2], [3, 1, 2]];
%e A291355 a(4) = 4: [[1, 3, 2, 4], [2, 4, 3, 1], [3, 1, 2, 4], [4, 2, 3, 1]];
%e A291355 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 A291355 (Ruby)
%o A291355 def search(a, sum, k, size, num)
%o A291355   if num == size + 1
%o A291355     @cnt += 1
%o A291355   else
%o A291355     (1..size).each{|i|
%o A291355       if a[i - 1] == 0 && (sum + i ** k) % num == 0
%o A291355         a[i - 1] = 1
%o A291355         search(a, sum + i ** k, k, size, num + 1)
%o A291355         a[i - 1] = 0
%o A291355       end
%o A291355     }
%o A291355   end
%o A291355 end
%o A291355 def A(k, n)
%o A291355   a = [0] * n
%o A291355   @cnt = 0
%o A291355   search(a, 0, k, n, 1)
%o A291355   @cnt
%o A291355 end
%o A291355 def A291355(n)
%o A291355   (0..n).map{|i| A(3, i)}
%o A291355 end
%o A291355 p A291355(20)
%Y A291355 Cf. A291445.
%K A291355 nonn
%O A291355 0,4
%A A291355 _Seiichi Manyama_, Aug 23 2017
%E A291355 a(0), a(13)-a(30) from _Alois P. Heinz_, Aug 23 2017
%E A291355 a(31)-a(55) from _Giovanni Resta_, Aug 23 2017