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

1,3

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

Links

Table of n, a(n) for n=1..52.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000290.

Sequence in context: A167300 A167347 A028604 * A337550 A088663 A291681

Adjacent sequences: A292528 A292529 A292530 * A292532 A292533 A292534

Keyword

nonn,easy

Author

J. Lowell, Sep 18 2017

Extensions

More terms from Giovanni Resta, Sep 19 2017

Status

approved