[seqfan] Re: Array A255483.
hv at crypt.org
hv at crypt.org
Sun Sep 18 11:21:42 CEST 2016
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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