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

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
(PARI) A276156(n) = { my(s=0, p=1, r=1); while(n, if(n%2, s += r); n>>=1; p = nextprime(1+p); r *= p); (s); }; \\ Antti Karttunen, Feb 03 2022

Crossrefs

Cf. A000040, A001511, A002110, A007088, A007814, A019565, A049345, A257993, A276084, A276085, A276154, A351073, A328461, A328473, A328474, A328571, A328831, A328836.

Subsequences: A328233, A328832, A328462 (odd bisection).

Fixed points of A328841, positions of zeros in A328842 and in A329032, positions of ones in A328581 and in A328582.

Cf. also table A328464 (and its rows).

Cf. also A059590, A283985, A290249, A342921.

Sequence in context: A257262 A293397 A059590 * A144705 A028733 A028789

Adjacent sequences: A276153 A276154 A276155 * A276157 A276158 A276159

Keyword

nonn,base

Author

Antti Karttunen, Aug 24 2016

Status

approved