A276156 - OEIS

A276156

Numbers obtained by reinterpreting base-2 representation of n in primorial base: a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1 + A276154(a(n)).

0, 1, 2, 3, 6, 7, 8, 9, 30, 31, 32, 33, 36, 37, 38, 39, 210, 211, 212, 213, 216, 217, 218, 219, 240, 241, 242, 243, 246, 247, 248, 249, 2310, 2311, 2312, 2313, 2316, 2317, 2318, 2319, 2340, 2341, 2342, 2343, 2346, 2347, 2348, 2349, 2520, 2521, 2522, 2523, 2526, 2527, 2528, 2529, 2550, 2551, 2552, 2553, 2556, 2557, 2558, 2559, 30030, 30031

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OFFSET

0,3

COMMENTS

Numbers that are sums of distinct primorial numbers, A002110.

Numbers with no digits larger than one in primorial base, A049345.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..8191

Index entries for sequences related to primorial base

FORMULA

a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1+A276154(a(n)).

Other identities. For all n >= 0:

a(n) = A276085(A019565(n)).

A049345(a(n)) = A007088(n).

A257993(a(n)) = A001511(n).

A276084(a(n)) = A007814(n).

A051903(a(n)) = A351073(n).

MATHEMATICA

nn = 65; b = MixedRadix[Reverse@ Prime@ Range[IntegerLength[nn, 2] - 1]]; Table[FromDigits[IntegerDigits[n, 2], b], {n, 0, 65}] (* Version 10.2, or *)

Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ IntegerDigits[n, 2], {n, 0, 65}] (* Michael De Vlieger, Aug 26 2016 *)

PROG

(Scheme, two versions)

;; Almost standalone, requiring only A000040:

(define (A276156 n) (let loop ((n n) (s 0) (pr 1) (i 1)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) s (* (A000040 i) pr) (+ 1 i))) (else (loop (/ (- n 1) 2) (+ s pr) (* (A000040 i) pr) (+ 1 i))))))

;; One using memoization-macro, implementing the given recurrence:

(definec (A276156 n) (cond ((zero? n) n) ((even? n) (A276154 (A276156 (/ n 2)))) (else (+ 1 (A276154 (A276156 (/ (- n 1) 2)))))))

(Python)

from sympy import prime, primorial, primepi, factorint

from operator import mul

def a002110(n): return 1 if n<1 else primorial(n)

def a276085(n):

f=factorint(n)

return sum([f[i]*a002110(primepi(i) - 1) for i in f])

def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) # after Chai Wah Wu

def a(n): return 0 if n==0 else a276085(a019565(n))

print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 23 2017