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Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.
4

%I #34 Aug 26 2017 14:06:53

%S 1,2,6,12,30,78,186,414,912,2064,4338,9798,20106,40974,80196,158322,

%T 309414,615558,1212402,2417136,4776654,9497508,18726708,37056150,

%U 72946116,144230640,284660874,564451830,1118803818,2224792026,4420041210,8791590168,17456783136

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

%C The permutation [1,...,n] satisfies the conditions since Sum_{i=1..n} i^3 = (Sum_{i=1..n})^2. Similarly, [n,...,1] satisfies the conditions since Sum_{i=m..n} i^3 = (Sum_{i=m..n} i)*(n*(n+1)+m*(m-1))/2. Thus a(n) >= 2 for n > 1 and a(n) is nondecreasing. Seems to approximately double for each successive n. - _Chai Wah Wu_, Aug 24 2017

%H Chai Wah Wu, <a href="/A291445/b291445.txt">Table of n, a(n) for n = 1..100</a>

%e 5 divides 5^3,

%e 5 + 4 divides 5^3 + 4^3,

%e 5 + 4 + 3 divides 5^3 + 4^3 + 3^3,

%e 5 + 4 + 3 + 2 divides 5^3 + 4^3 + 3^3 + 2^3,

%e 5 + 4 + 3 + 2 + 1 divides 5^3 + 4^3 + 3^3 + 2^3 + 1^3.

%e So [5, 4, 3, 2, 1] satisfies all the conditions.

%e 1 divides 1^3,

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

%e 1 + 2 + 6 divides 1^3 + 2^3 + 6^3,

%e 1 + 2 + 6 + 5 divides 1^3 + 2^3 + 6^3 + 5^3,

%e 1 + 2 + 6 + 5 + 4 divides 1^3 + 2^3 + 6^3 + 5^3 + 4^3,

%e 1 + 2 + 6 + 5 + 4 + 3 divides 1^3 + 2^3 + 6^3 + 5^3 + 4^3 + 3^3.

%e So [1, 2, 6, 5, 4, 3] satisfies all the conditions.

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

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

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

%e a(3) = 6: [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]];

%e a(4) = 12: [[1, 2, 3, 4], [1, 3, 2, 4], [2, 1, 3, 4], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 3, 1], [3, 1, 2, 4], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 2, 1], [4, 2, 3, 1], [4, 3, 2, 1]];

%e a(5) = 30: [[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, 3, 4, 5, 1], [2, 3, 5, 4, 1], [2, 4, 3, 1, 5], [2, 4, 3, 5, 1], [2, 5, 3, 4, 1], [3, 1, 2, 4, 5], [3, 2, 1, 4, 5], [3, 2, 4, 1, 5], [3, 2, 4, 5, 1], [3, 2, 5, 4, 1], [3, 4, 2, 1, 5], [3, 4, 2, 5, 1], [3, 4, 5, 2, 1], [3, 5, 2, 4, 1], [3, 5, 4, 2, 1], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [4, 3, 2, 1, 5], [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]].

%Y Cf. A067957, A291355, A291518, A291519.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Aug 23 2017

%E a(13)-a(33) from _Chai Wah Wu_, Aug 24 2017