A278159 - OEIS

A278159

Run length transform of primorials, A002110.

1, 2, 2, 6, 2, 4, 6, 30, 2, 4, 4, 12, 6, 12, 30, 210, 2, 4, 4, 12, 4, 8, 12, 60, 6, 12, 12, 36, 30, 60, 210, 2310, 2, 4, 4, 12, 4, 8, 12, 60, 4, 8, 8, 24, 12, 24, 60, 420, 6, 12, 12, 36, 12, 24, 36, 180, 30, 60, 60, 180, 210, 420, 2310, 30030, 2, 4, 4, 12, 4, 8, 12, 60, 4, 8, 8, 24, 12, 24, 60, 420, 4, 8, 8, 24, 8, 16, 24, 120, 12, 24, 24, 72, 60, 120, 420

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OFFSET

0,2

COMMENTS

Like every run length transform this sequence satisfies for all i, j: A278222(i) = A278222(j) => a(i) = a(j).

LINKS

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

Index entries for sequences related to binary expansion of n

Index entries for sequences computed with run length transform

FORMULA

a(n) = A124859(A005940(1+n)).

EXAMPLE

For n=7, "111" in binary, there is a run of 1-bits of length 3, thus a(7) = product of A002110(3), = A002110(3) = 30.

For n=39, "10111" in binary, there are two runs, of lengths 1 and 3, thus a(39) = A002110(1) * A002110(3) = 2*30 = 60.

MATHEMATICA

f[n_] := Product[Prime[k], {k, 1, n}]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 94}] (* Jean-François Alcover, Jul 11 2017 *)

PROG

(Scheme)

(define (A278159 n) (fold-left (lambda (a r) (* a (A002110 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))

;; See A227349 for the required other functions.

(Python)

from math import prod

from re import split

from sympy import primorial

def RLT(n, f):

""" run length transform of a function f """

return prod(f(len(d)) for d in split('0+', bin(n)[2:]) if d != '') if n > 0 else 1

def A278159(n): return RLT(n, primorial) # Chai Wah Wu, Feb 04 2022