[seqfan] Re: Array A255483.
M. F. Hasler
oeis at hasler.fr
Sun Sep 18 15:43:29 CEST 2016
Another characterization of A255483 <http://oeis.org/A255483> is the
following:
The first column is A123098: multiplicative encoding of Pascal's triangle
mod 2 (e.g. n=2 => 1 0 1 => 2^1 * 3^0 * 5^1),
and subsequent columns are obtained by applying the function A003961, i.e.,
replacing each prime factor by the next larger prime. (Equivalently, the
n-th line of the binary Pascal triangle is shifted by inserting an initial
0.)
This also shows that the sequence is injective and with squarefree terms,
and allows to compute any term directly via the corresponding binomial
coefficients in Z_2.
- Maximilian
On Sun, Sep 18, 2016 at 5:21 AM, <hv at crypt.org> wrote:
> Antti Karttunen <antti.karttunen at gmail.com> wrote:
> :This interesting array
> :http://oeis.org/A255483
> :seems to be injective.
> :
> :Is there a simple way to characterize the terms that are present?
>
> I think so: there is no opportunity for the prime factorization of any
> entry to acquire a squared term, so let c_i be 1 when a(n) is divisible
> by the i'th prime, and 0 otherwise; we can then replace:
> a(n) = p_0 ^ c_0 * p_1 ^ c_1 * ... p_{k-1} ^ c_{k-1}
> with:
> b(n) = c_0 * 2^0 + c_1 * 2^1 + ... c_{k-1} * 2^{k-1}
>
> The operation on the transformed b(n) is just exclusive-or, with an initial
> row of 2^n:
> 1 2 4 8 ...
> 3 6 12 24 ...
> 5 10 20 40 ...
> 15 30 60 120 ...
> 17 34 68 136 ...
>
> Each term is always double the one to its left, and the first column has:
> a(1) = 1; a(n+1) = a(n) XOR (2 a(n))
> .. which looks to be A001317.
>
> Hugo
>
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>
--
Maximilian
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