A365406 Numbers j whose largest divisor <= sqrt(j) is a power of 2.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 122, 124, 127, 128, 131, 134, 136, 137

Offset: 1

Author

Omar E. Pol, Oct 10 2023

Keywords

• nonn

Comments

Also indices of the powers of 2 in A033676.

Also numbers in increasing order from the columns k of A163280 where k is a power of 2.

Observation: at least the first 82 terms of the subsequence of terms with no middle divisors (that is 3, 5, 7, 10, ...) coincide with at least the first 82 terms of A246955.

For the definition of middle divisor see A067742.

From Peter Munn, Oct 26 2023: (Start)

Most of the early terms are in A342081, which consists of powers of 2 together with products of a prime and a power of 2 where the prime is the larger. The exceptions are 24, 72, 80, 96, 112, ... .

The odd terms clearly consist of 1 and the odd primes. We can fully characterize the even terms by their A290110 values, which depend on the relative sizes of a number's divisors. A290110 provides a refinement of the classification of numbers by prime signature (cf. A212171): see the example below for numbers with the same prime signature as 48.

(End)

Examples

			From _Peter Munn_, Oct 26 2023: (Start)
The table below looks at numbers j with prime signature (4, 1), showing the presence of j and its characterization by A290110(j):
    j             A290110(j)  present
    48 = 2^4 * 3      16         no
    80 = 2^4 * 5      21        yes
   112 = 2^4 * 7      21        yes
   162 = 2 * 3^4      36         no
   176 = 2^4 * 11     38         no
   208 = 2^4 * 13     38         no
   272 = 2^4 * 17     51        yes
   304 = 2^4 * 19     51        yes
   368 = 2^4 * 23     51        yes
  ...
Clearly any odd composite number is exempted, for example:
   891 = 3^4 * 11     21         no
  6723 = 3^4 * 83     51         no
Note that A290110(j) = 36 for j = 2 * p^4, prime p; and A290110(j) = 51 for j = 2^4 * p, prime p >= 17.
(End)
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A342081 (a subsequence), A365408 (complement), A365716 (characteristic function).

Cf. A033676, A067742, A071561, A163280, A212171, A246955, A290110.

Programs

Mathematica

q[n_] := Module[{d = Divisors[n], mid}, mid = d[[Ceiling[Length[d]/2]]]; mid == 2^IntegerExponent[mid, 2]]; Select[Range[150], q] (* Amiram Eldar, Oct 11 2023 *)

PARI

f(n) = local(d); if(n<2, 1, d=divisors(n); d[(length(d)+1)\2]); \\ A033676
isp2(n) = 2^logint(n,2) == n;
isok(k) = isp2(f(k)); \\ Michel Marcus, Oct 11 2023

Python

from itertools import count, islice
from sympy import divisors
def A365406_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda i:(a:=(d:=divisors(i))[len(d)-1>>1])==1<A365406_list = list(islice(A365406_gen(),30)) # Chai Wah Wu, Oct 18 2023