A291519 -id:A291519 - cp's OEIS frontend

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

Seiichi Manyama, Aug 23 2017

nonn

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

			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]].

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

Cf. A067957, A291355, A291518, A291519.

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

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.

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