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A286575 Run-length transform of A001316. 5
1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 8, 4, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4, 8, 8, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 2, 4, 4, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 8, 16, 4, 8, 8, 8, 8, 16, 4, 8, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16384

Index entries for sequences related to binary expansion of n

Index entries for sequences computed with run length transform

FORMULA

a(n) = A037445(A005940(1+n)).

a(n) = A000079(A286574(n)).

EXAMPLE

For n = 0, there are no 1-runs, and thus a(0) = 1 as an empty product.

For n = 29, "11101" in binary, there are two 1-runs, of lengths 1 and 3, thus a(29) = A001316(1) * A001316(3) = 2*4 = 8.

MATHEMATICA

Table[Times @@ Map[Sum[Mod[#, 2] &@ Binomial[#, k], {k, 0, #}] &@ Length@ # &, DeleteCases[Split@ IntegerDigits[n, 2], _?(First@ # == 0 &)]], {n, 0, 108}] (* Michael De Vlieger, May 29 2017 *)

PROG

(Scheme)

(define (A286575 n) (fold-left (lambda (a r) (* a (A001316 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))

(define (bisect lista parity) (let loop ((lista lista) (i 0) (z (list))) (cond ((null? lista) (reverse! z)) ((eq? i parity) (loop (cdr lista) (modulo (1+ i) 2) (cons (car lista) z))) (else (loop (cdr lista) (modulo (1+ i) 2) z)))))

(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))

(define (A001316 n) (let loop ((n n) (z 1)) (cond ((zero? n) z) ((even? n) (loop (/ n 2) z)) (else (loop (/ (- n 1) 2) (* z 2))))))

(Python)

from sympy import factorint, prime, log

import math

def wt(n): return bin(n).count("1")

def a037445(n):

    f=factorint(n)

    return 2**sum([wt(f[i]) for i in f])

def A(n): return n - 2**int(math.floor(log(n, 2)))

def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))

def a(n): return a037445(b(n)) # Indranil Ghosh, May 30 2017

CROSSREFS

Cf. A000079, A001316, A005940, A037445, A227349, A247282, A286574.

Sequence in context: A037445 A286324 A186643 * A270438 A003036 A089818

Adjacent sequences:  A286572 A286573 A286574 * A286576 A286577 A286578

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, May 28 2017

STATUS

approved

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Last modified February 20 14:37 EST 2018. Contains 299380 sequences. (Running on oeis4.)