cp's OEIS Frontend

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.

%I A110765 #39 Aug 30 2025 15:57:18
%S A110765 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,
%T A110765 66,42,462,30,330,210,2310,2,26,22,286,14,182,154,2002,10,130,110,
%U A110765 1430,70,910,770,10010,6,78,66,858,42,546,462,6006,30,390,330,4290,210,2730
%N 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.
%C A110765 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
%C A110765 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
%H A110765 Peter Munn, <a href="/A110765/b110765.txt">Table of n, a(n) for n = 0..10000</a> (essentially from Reinhard Zumkeller)
%F A110765 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
%e A110765 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.
%e A110765 n = 10: binary(10) = 1010, so a(10) = 2^1 * 3^0 * 5^1 * 7^0 = 2*1*5*1 = 10.
%t A110765 Array[Times @@ Prime@ Flatten@ Position[#, 1] &@ IntegerDigits[#, 2] &, 61] (* _Michael De Vlieger_, Feb 28 2021 *)
%o A110765 (PARI) a(n)=factorback(Mat(vector(#n=binary(n),j,[prime(j),n[j]])~))
%o A110765 (PARI) a(n)=prod(j=1,#n=binary(n),prime(j)^n[j])  \\ _M. F. Hasler_, Mar 25 2011
%o A110765 (Haskell)
%o A110765 a110765 = product . zipWith (^) a000040_list .  reverse . a030308_row
%o A110765 -- _Reinhard Zumkeller_, Aug 28 2014
%o A110765 (Python)
%o A110765 from sympy import prime
%o A110765 from operator import mul
%o A110765 from functools import reduce
%o A110765 def A110765(n):
%o A110765     return reduce(mul, (prime(i) for i,d in enumerate(bin(n)[2:],start=1) if int(d)))
%o A110765 # _Chai Wah Wu_, Sep 05 2014
%o A110765 (Python)
%o A110765 # implementation using recursion
%o A110765 from sympy import prime
%o A110765 def _A110765(n):
%o A110765     nlen = len(n)
%o A110765     return _A110765(n[:-1])*(prime(nlen) if int(n[-1]) else 1) if nlen > 1 else int(n) + 1
%o A110765 def A110765(n):
%o A110765     return _A110765(bin(n)[2:])
%o A110765 # _Chai Wah Wu_, Sep 05 2014
%Y A110765 Cf. A110766.
%Y A110765 Cf. A030308, A000040, A000523, A019565.
%Y A110765 Range of terms: {1} U A039956.
%K A110765 base,easy,nonn,look,changed
%O A110765 0,2
%A A110765 _Amarnath Murthy_, Aug 12 2005
%E A110765 More terms from Stacy Hawthorne (shawtho1(AT)ashland.edu), Oct 31 2005
%E A110765 Name edited by _Peter Munn_, May 28 2025
%E A110765 a(0) prefixed by _Peter Munn_, Aug 26 2025