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

Author

Antti Karttunen, Aug 24 2016

Keywords

• nonn
• base

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

Crossrefs

Cf. A000040, A001511, A002110, A007088, A007814, A019565, A049345, A257993, A276084, A276085, A276154, A351073, A328461, A328473, A328474, A328571, A328831, A328836, A341518, A344591.

Complement of A177711.

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

Conjectured subsequences: A328110, A380527.

Fixed points of A328841, positions of zeros in A328828, A328842, and A329032, positions of ones in A328581, A328582, and A381032.

Positions of terms < 2 in A328114.

Indices where A327860 and A329029 coincide.

Cf. also table A328464 (and its rows).

Cf. also A059590, A283985, A290249, A342921.

Programs

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 *)

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

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

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).