A119389 Numerator of (1^2/n + 2^2/(n-1) + ... + k^2/(n-k+1) + ... + (n-1)^2/2 + n^2/1).

1, 9, 34, 265, 186, 1141, 2868, 31401, 18635, 477301, 91192, 8051069, 4508441, 3336145, 22048024, 410111791, 223063947, 3057889621, 823596665, 706952715, 125961187, 6173866701, 9838037952, 521135614075, 275363139571

Offset

1,2

Comments

p divides a(p-1) for prime p>2. p divides a(2p-1) for all prime p. p divides a(3p-1) for all prime p. p divides a(4p-1) for all prime p except p=3. p divides a(5p-1) for prime p>3. p divides a(6p-1) for all prime except p=5. . p^2 divides a(p^2-1) for prime p>2. p^2 divides a(2p^2-1) for all prime p. p^2 divides a(3p^2-1) for all prime p. . p^3 divides a(p^3-1) for prime p>2. . p^k divides a(p^k-1) for prime p>2 and integer k>1. p^k divides a(m*p^k-1) for all prime p and integer m,k>1.

Links

Table of n, a(n) for n=1..25.

Formula

a(n) = Numerator[Sum[k^2/(n-k+1),{k,1,n}]]. a(n) = Numerator[HarmonicNumber[n]*(n+1)^2 - 3*n(n+1)/2]. a(n) = Numerator[A001008[n]/A002805[n]*(n+1)^2 - 3*A000217[n]].

Mathematica

Numerator[Table[Sum[k^2/(n-k+1), {k, 1, n}], {n, 1, 50}]]

Crossrefs

Cf. A027612, A001008, A002805, A000217.

Sequence in context: A050478 A204426 A154393 * A197273 A067960 A119757

Adjacent sequences: A119386 A119387 A119388 * A119390 A119391 A119392

Keyword

frac,nonn

Author

Alexander Adamchuk, Jul 26 2006

Status

approved