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a(n) = Product_{i in row n of A245562} prime(i).
6

%I #43 Mar 19 2023 23:36:06

%S 1,2,2,3,2,4,3,5,2,4,4,6,3,6,5,7,2,4,4,6,4,8,6,10,3,6,6,9,5,10,7,11,2,

%T 4,4,6,4,8,6,10,4,8,8,12,6,12,10,14,3,6,6,9,6,12,9,15,5,10,10,15,7,14,

%U 11,13,2,4,4,6,4,8,6,10,4,8,8,12,6,12,10,14,4,8,8,12,8,16,12,20,6,12,12,18

%N a(n) = Product_{i in row n of A245562} prime(i).

%C This is the Run Length Transform of S(n) = {1,2,3,5,7,11,...} (1 followed by the primes).

%C The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

%H Alois P. Heinz, <a href="/A246029/b246029.txt">Table of n, a(n) for n = 0..8191</a>

%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.

%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.

%H N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = A181819(A005940(1+n)). - _Antti Karttunen_, Oct 15 2016

%e From _Omar E. Pol_, Feb 12 2015: (Start)

%e Written as an irregular triangle in which row lengths is A011782:

%e 1;

%e 2;

%e 2,3;

%e 2,4,3,5;

%e 2,4,4,6,3,6,5,7;

%e 2,4,4,6,4,8,6,10,3,6,6,9,5,10,7,11;

%e 2,4,4,6,4,8,6,10,4,8,8,12,6,12,10,14,3,6,6,9,6,12,9,15,5,10,10,15,7,14,11,13;

%e ...

%e Right border gives the noncomposite numbers. This is simply a restatement of the theorem that this sequence is the Run Length Transform of A008578.

%e (End)

%p ans:=[];

%p for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;

%p for i from 1 to L1 do

%p if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;

%p elif out1 = 0 and t1[i] = 1 then c:=c+1;

%p elif out1 = 1 and t1[i] = 0 then c:=c;

%p elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;

%p fi;

%p if i = L1 and c>0 then lis:=[c,op(lis)]; fi;

%p od:

%p a:=mul(ithprime(i), i in lis);

%p ans:=[op(ans),a];

%p od:

%p ans;

%t f[n_, i_, x_] := f[n, i, x] = Which[n == 0, x, EvenQ[n], f[n/2, i + 1, x], True, f[(n - 1)/2, i, x*Prime[i]]];

%t a5940[n_] := f[n - 1, 1, 1];

%t a181819[n_] := Times @@ Prime[FactorInteger[n][[All, 2]]];

%t a[0] = 1; a[n_] := a181819[a5940[n + 1]];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 19 2018, after _Antti Karttunen_ *)

%o (Python)

%o from operator import mul

%o from functools import reduce

%o from re import split

%o from sympy import prime

%o def A246029(n):

%o return reduce(mul,(prime(len(d)) for d in split('0+',bin(n)[2:]) if d != '')) if n > 0 else 1

%o # _Chai Wah Wu_, Sep 12 2014

%Y Cf. A245562, A000045, A001045, A005940, A008578, A011782, A071053, A181819, A245565, A246028, A253081 (partial sums).

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Aug 15 2014; revised Sep 05 2014