cp's OEIS Frontend

A023534 Numbers k such that the largest power of 2 dividing k equals 2^omega(k).

%I A023534 #35 Aug 15 2025 12:49:50
%S A023534 1,2,12,20,28,36,44,52,68,76,92,100,108,116,120,124,148,164,168,172,
%T A023534 188,196,212,236,244,264,268,280,284,292,312,316,324,332,356,360,388,
%U A023534 404,408,412,428,436,440,452,456,484,500,504,508,520,524,548,552,556
%N A023534 Numbers k such that the largest power of 2 dividing k equals 2^omega(k).
%C A023534 That is, numbers k such that A001221(k) is equal to A007814(k), where A001221(k) = omega(k) is the number of distinct primes dividing k, and A007814(k) is the exponent of the highest power of 2 dividing k.
%C A023534 Numbers k that are unitarily divisible by the number of unitary divisors of k (A034444), i.e., numbers k such that A034444(k) is a unitary divisor of k. Subsequence of A048166. - _Amiram Eldar_, Aug 15 2025
%H A023534 Amiram Eldar, <a href="/A023534/b023534.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from G. C. Greubel)
%e A023534 omega(12) = 2 and 4 = 2^2 is the largest power of 2 dividing 12, hence 12 is in the sequence.
%t A023534 Select[Range[600],IntegerExponent[#,2]==PrimeNu[#]&] (* _Harvey P. Dale_, Jun 26 2011 *)
%o A023534 (PARI) isok(n) = omega(n) == valuation(n, 2); \\ _Michel Marcus_, Apr 16 2015
%o A023534 (Python)
%o A023534 from itertools import count, islice
%o A023534 from sympy.ntheory.factor_ import primenu
%o A023534 def A023534_gen(startvalue=1): # generator of terms
%o A023534     return filter(lambda n:(~n & n-1).bit_length() == primenu(n),count(max(startvalue,1)))
%o A023534 A023534_list = list(islice(A023534_gen(),30)) # _Chai Wah Wu_, Jul 05 2022
%Y A023534 Cf. A039700, A001221 (omega), A007814 (2-adic valuation), A034444, A048166.
%K A023534 nonn
%O A023534 1,2
%A A023534 _Benoit Cloitre_, Sep 04 2002