A110765 Let n in binary be a k-bit number with bits d_1, d_2, ..., d_k (concatenated). a(n) = 2^d_1 * 3^d_2 * ... * prime(k)^d_k, where prime(k) is the k-th prime.

1, 2, 2, 6, 2, 10, 6, 30, 2, 14, 10, 70, 6, 42, 30, 210, 2, 22, 14, 154, 10, 110, 70, 770, 6, 66, 42, 462, 30, 330, 210, 2310, 2, 26, 22, 286, 14, 182, 154, 2002, 10, 130, 110, 1430, 70, 910, 770, 10010, 6, 78, 66, 858, 42, 546, 462, 6006, 30, 390, 330, 4290, 210, 2730

Offset: 0

Author

Amarnath Murthy, Aug 12 2005

Keywords

• base
• easy
• nonn
• look
• changed

Comments

All terms after a(0) have 2-adic valuation equal to 1, i.e., they equal twice an odd (and squarefree) number, since the first digit in base two will always be "1". - M. F. Hasler, Mar 25 2011

2 appears at index n = 2^k for k >= 0, since such n_2 begins with "1" followed by k zeros, and 2^1 * 3^0 * ... * p_(k+1)^0 = 2. - Michael De Vlieger, Feb 28 2021

Examples

			n = 7: binary(7) = 111, and the first three primes are 2, 3, 5, so a(7) = 2^1 * 3^1 * 5^1 = 2*3*5 = 30.
n = 10: binary(10) = 1010, so a(10) = 2^1 * 3^0 * 5^1 * 7^0 = 2*1*5*1 = 10.
		

Links

• Peter Munn, Table of n, a(n) for n = 0..10000 (essentially from Reinhard Zumkeller)

Crossrefs

Cf. A110766.

Cf. A030308, A000040, A000523, A019565.

Range of terms: {1} U A039956.

Programs

Haskell

a110765 = product . zipWith (^) a000040_list .  reverse . a030308_row
-- Reinhard Zumkeller, Aug 28 2014

Mathematica

Array[Times @@ Prime@ Flatten@ Position[#, 1] &@ IntegerDigits[#, 2] &, 61] (* Michael De Vlieger, Feb 28 2021 *)

PARI

a(n)=factorback(Mat(vector(#n=binary(n),j,[prime(j),n[j]])~))

PARI

a(n)=prod(j=1,#n=binary(n),prime(j)^n[j])  \\ M. F. Hasler, Mar 25 2011

Python

from sympy import prime
from operator import mul
from functools import reduce
def A110765(n):
    return reduce(mul, (prime(i) for i,d in enumerate(bin(n)[2:],start=1) if int(d)))
# Chai Wah Wu, Sep 05 2014

Python

# implementation using recursion
from sympy import prime
def _A110765(n):
    nlen = len(n)
    return _A110765(n[:-1])*(prime(nlen) if int(n[-1]) else 1) if nlen > 1 else int(n) + 1
def A110765(n):
    return _A110765(bin(n)[2:])
# Chai Wah Wu, Sep 05 2014

Formula

a(0) = 1; a(2n) = a(n); a(2n+1) = a(n) * A000040(1+A000523(2n+1)), where A000040(k) is the k-th prime and A000523(k) = floor(log_2(k)) . - Peter Munn, Aug 26 2025

Extensions

More terms from Stacy Hawthorne (shawtho1(AT)ashland.edu), Oct 31 2005

Name edited by Peter Munn, May 28 2025

a(0) prefixed by Peter Munn, Aug 26 2025