A292531 a(n) = 0 if n is a power of 2. Otherwise, product of 2 numbers nearest n that have more 2's in their prime factorization than n.

0, 0, 8, 0, 24, 32, 48, 0, 80, 96, 120, 128, 168, 192, 224, 0, 288, 320, 360, 384, 440, 480, 528, 512, 624, 672, 728, 768, 840, 896, 960, 0, 1088, 1152, 1224, 1280, 1368, 1440, 1520, 1536, 1680, 1760, 1848, 1920, 2024, 2112, 2208, 2048, 2400, 2496, 2600, 2688

Offset: 1

Author

J. Lowell, Sep 18 2017

Keywords

• nonn
• easy

Comments

1) For all odd n, a(n) = n^2 - 1.

2) All numbers in sequence are divisible by 8.

3) a(n) is not divisible by 16 if and only if n = 8k+3 or n = 8k+5.

Proposition 2) is true. Proof: 0 mod 2 = 0, so the conjecture trivially holds when n is a power of 2. For n not a power of 2, a(n) has by definition the repeated prime factor 2^2 and so is divisible by 8 when a(n) > 4. - Felix Fröhlich, Sep 19 2017

Examples

			a(40) = 1536 because 40 has three 2's in its prime factorization, and the closest integers to 40 that have at least four 2's are 32 and 48, and 32 times 48 = 1536.
		

Crossrefs

Cf. A000290.

Programs

Mathematica

a[n_] := Block[{p = 2 2^IntegerExponent[n, 2]}, Floor[n/p] Ceiling[n/p] p^2]; Array[a, 60] (* Giovanni Resta, Sep 19 2017 *)

Formula

a(n) = p^2 * ceiling(n/p) * floor(n/p), where p = A171977(n). - Giovanni Resta, Sep 19 2017

Extensions

More terms from Giovanni Resta, Sep 19 2017