

A246685


Run Length Transform of sequence 1, 3, 5, 17, 257, 65537, ... (1 followed by Fermat numbers).


3



1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 17, 1, 1, 1, 3, 1, 1, 3, 5, 3, 3, 3, 9, 5, 5, 17, 257, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 17, 3, 3, 3, 9, 3, 3, 9, 15, 5, 5, 5, 15, 17, 17, 257, 65537, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 17, 1, 1, 1, 3, 1, 1, 3, 5, 3, 3, 3, 9, 5, 5, 17, 257
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

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).
This sequence is obtained by applying Run Length Transform to sequence b = 1, 3, 5, 17, 257, 65537, ... (1 followed by Fermat numbers, with b(1) = 1, b(2) = 3, b(3) = 5, ..., b(n) = 2^(2^(n2)) + 1 for n >= 2).


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..1024


EXAMPLE

115 is '1110011' in binary. The run lengths of 1runs are 2 and 3, thus we multiply the second and the third elements of the sequence 1, 3, 5, 17, 257, 65537, ... to get a(115) = 3*5 = 15.


MATHEMATICA

f[n_] := Switch[n, 01, 1, _, 2^(2^(n2))+1]; Table[Times @@ (f[Length[#]] &) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 95}] (* JeanFranÃ§ois Alcover, Jul 11 2017 *)


PROG

(MIT/GNU Scheme)
(define (A246685 n) (foldleft (lambda (a r) (if (= 1 r) a (* a (A000215 ( r 2))))) 1 (bisect (reverse (binexp>runcount1list n)) ( 1 (modulo n 2)))))
(define (A000215 n) (+ 1 (A000079 (A000079 n))))
(define (A000079 n) (expt 2 n))
;; Other functions as in A227349.
(Python)
# use RLT function from A278159
def A246685(n): return RLT(n, lambda m: 1 if m <= 1 else 2**(2**(m2))+1) # Chai Wah Wu, Feb 04 2022


CROSSREFS

Cf. A003714 (gives the positions of ones).
Cf. A000215.
A001316 is obtained when the same transformation is applied to A000079, the powers of two. Cf. also A001317.
Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246674, A247282.
Sequence in context: A027960 A319182 A247282 * A218618 A271451 A131248
Adjacent sequences: A246682 A246683 A246684 * A246686 A246687 A246688


KEYWORD

nonn


AUTHOR

Antti Karttunen, Sep 22 2014


STATUS

approved



