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A246029 a(n) = Product_{i in row n of A245562} prime(i). 5
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..8191

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.

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: Part 1, Part 2

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

Index entries for sequences related to cellular automata

FORMULA

a(n) = A181819(A005940(1+n)). - Antti Karttunen, Oct 15 2016

EXAMPLE

From Omar E. Pol, Feb 12 2015: (Start)

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

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;

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

...

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

(End)

MAPLE

ans:=[];

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

for i from 1 to L1 do

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

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

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

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

fi;

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

od:

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

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

od:

ans;

MATHEMATICA

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]]];

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

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

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

Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Aug 19 2018, after Antti Karttunen *)

PROG

(Python)

from operator import mul

from functools import reduce

from re import split

from sympy import prime

def A246029(n):

return reduce(mul, (prime(len(d)) for d in split('0+', bin(n)[2:]) if d != '')) if n > 0 else 1 # Chai Wah Wu, Sep 12 2014

CROSSREFS

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

Sequence in context: A286532 A286622 A286589 * A245564 A214126 A205378

Adjacent sequences:  A246026 A246027 A246028 * A246030 A246031 A246032

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 15 2014; revised Sep 05 2014

STATUS

approved

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Last modified January 20 12:50 EST 2019. Contains 319330 sequences. (Running on oeis4.)