A291445 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.

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, 17456783136

Offset

1,2

Comments

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

Links

Chai Wah Wu, Table of n, a(n) for n = 1..100

Example

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

Crossrefs

Cf. A067957, A291355, A291518, A291519.

Sequence in context: A032177 A095349 A291518 * A320664 A022916 A073949

Adjacent sequences: A291442 A291443 A291444 * A291446 A291447 A291448

Keyword

nonn

Author

Seiichi Manyama, Aug 23 2017

Extensions

a(13)-a(33) from Chai Wah Wu, Aug 24 2017

Status

approved