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

0,2

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

Links

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

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

From Peter Munn, Sep 05 2025: (Start)

a(n) = A019565(A030101(n)).

a(n) = Product_{k=1..A029837(n+1)} A000040(k)^A030308(n, k+c-1), where c is the column offset of A030308.

(End)

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A110766.

See the formula section for the relationships with A000040, A000523, A019565, A029837, A030101, A030308.

Range of terms: {1} U A039956.

Sequence in context: A099985 A298299 A352597 * A176991 A264666 A283362

Adjacent sequences: A110762 A110763 A110764 * A110766 A110767 A110768

Keyword

base,easy,nonn,look

Author

Amarnath Murthy, Aug 12 2005

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

Status

approved