A246685 - OEIS

%I #20 Feb 04 2022 14:36:43

%S 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,

%T 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,

%U 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

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

%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).

%C 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^(n-2)) + 1 for n >= 2).

%H Antti Karttunen, <a href="/A246685/b246685.txt">Table of n, a(n) for n = 0..1024</a>

%e 115 is '1110011' in binary. The run lengths of 1-runs 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.

%t f[n_] := Switch[n, 0|1, 1, _, 2^(2^(n-2))+1]; Table[Times @@ (f[Length[#]] &) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 95}] (* _Jean-François Alcover_, Jul 11 2017 *)

%o (MIT/GNU Scheme)

%o (define (A246685 n) (fold-left (lambda (a r) (if (= 1 r) a (* a (A000215 (- r 2))))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))

%o (define (A000215 n) (+ 1 (A000079 (A000079 n))))

%o (define (A000079 n) (expt 2 n))

%o ;; Other functions as in A227349.

%o (Python)

%o # use RLT function from A278159

%o def A246685(n): return RLT(n,lambda m: 1 if m <= 1 else 2**(2**(m-2))+1) # _Chai Wah Wu_, Feb 04 2022

%Y Cf. A003714 (gives the positions of ones).

%Y Cf. A000215.

%Y A001316 is obtained when the same transformation is applied to A000079, the powers of two. Cf. also A001317.

%Y Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246674, A247282.

%K nonn

%O 0,4

%A _Antti Karttunen_, Sep 22 2014