A246674 Run Length Transform of A000225.

1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 3, 7, 15, 1, 1, 1, 3, 1, 1, 3, 7, 3, 3, 3, 9, 7, 7, 15, 31, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 3, 7, 15, 3, 3, 3, 9, 3, 3, 9, 21, 7, 7, 7, 21, 15, 15, 31, 63, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 3, 7, 15, 1, 1, 1, 3, 1, 1, 3, 7, 3, 3, 3, 9, 7, 7, 15, 31, 3, 3, 3, 9, 3, 3, 9, 21, 3, 3, 3, 9, 9, 9, 21, 45, 7, 7, 7, 21, 7, 7, 21, 49, 15, 15, 15, 45, 31, 31, 63, 127, 1

Offset

0,4

Comments

a(n) can be also computed by replacing all consecutive runs of zeros in the binary expansion of n with * (multiplication sign), and then performing that multiplication, still in binary, after which the result is converted into decimal. See the example below.

Links

Antti Karttunen, Table of n, a(n) for n = 0..8192

Formula

For all n >= 0, a(A051179(n)) = A247282(A051179(n)) = A051179(n).

Example

115 is '1110011' in binary. The run lengths of 1-runs are 2 and 3, thus a(115) = A000225(2) * A000225(3) = ((2^2)-1) * ((2^3)-1) = 3*7 = 21.
The same result can be also obtained more directly, by realizing that '111' and '11' are the binary representations of 7 and 3, and 7*3 = 21.
From Omar E. Pol, Feb 15 2015: (Start)
Written as an irregular triangle in which row lengths are the terms of A011782:
1;
1;
1,3;
1,1,3,7;
1,1,1,3,3,3,7,15;
1,1,1,3,1,1,3,7,3,3,3,9,7,7,15,31;
1,1,1,3,1,1,3,7,1,1,1,3,3,3,7,15,3,3,3,9,3,3,9,21,7,7,7,21,15,15,31,63;
...
Right border gives 1 together with the positive terms of A000225.
(End)

Mathematica

f[n_] := 2^n - 1; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 100}] (* Jean-François Alcover, Jul 11 2017 *)

Prog

(MIT/GNU Scheme)
(define (A246674 n) (fold-left (lambda (a r) (* a (A000225 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
(define (A000225 n) (- (A000079 n) 1))
(define (A000079 n) (expt 2 n))
;; Other functions as in A227349.
(Python)
# uses RLT function in A278159
def A246674(n): return RLT(n, lambda m: 2**m-1) # Chai Wah Wu, Feb 04 2022

Crossrefs

Cf. A003714 (gives the positions of ones).

Cf. A000225, A051179.

A001316 is obtained when the same transformation is applied to A000079, the powers of two.

Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246685, A247282.

Sequence in context: A121745 A252983 A089312 * A363340 A058735 A107294

Adjacent sequences: A246671 A246672 A246673 * A246675 A246676 A246677

Keyword

nonn,base

Author

Antti Karttunen, Sep 08 2014

Status

approved