A365809 Odd numbers k such that A163511(k) is a square.

5, 11, 17, 23, 35, 41, 47, 65, 71, 83, 89, 95, 131, 137, 143, 161, 167, 179, 185, 191, 257, 263, 275, 281, 287, 323, 329, 335, 353, 359, 371, 377, 383, 515, 521, 527, 545, 551, 563, 569, 575, 641, 647, 659, 665, 671, 707, 713, 719, 737, 743, 755, 761, 767, 1025, 1031, 1043, 1049, 1055, 1091, 1097, 1103, 1121

Offset: 1

Author

Antti Karttunen, Oct 01 2023

Keywords

• nonn

Comments

The sequence is defined inductively as follows:

(a) it contains all numbers of the form 2^(2e) + 1, e >= 1,

and

(b) for any term a(n) and any e >= 0, (2^(2e+1) * a(n)) + 1 is also included as a term. There are no other kind of terms.

Contains no squares. See A365808 for a proof.

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Odd terms of A365808.

Cf. A016754, A052539 (subsequence without its initial 2), A163511.

Programs

PARI

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA365809(n) = (n%2 && issquare(A163511(n)));

PARI

A209229(n) = (n && !bitand(n,n-1));
isA365809(n) = if(n<=2,0,my(v=valuation(n-1,2)); if(A209229(n-1),!(v%2),(v%2)&&isA365809(n>>v)));

Python

from itertools import count, islice
def A365809_gen(): # generator of terms
    return map(lambda n:(3*(n+1)>>1)-1,filter(lambda n:n&3==3 and not '00' in bin(n),count(1)))
A365809_list = list(islice(A365809_gen(),63)) # Chai Wah Wu, Feb 12 2025