A365808 Numbers k such that A163511(k) is a square.

0, 2, 5, 8, 11, 17, 20, 23, 32, 35, 41, 44, 47, 65, 68, 71, 80, 83, 89, 92, 95, 128, 131, 137, 140, 143, 161, 164, 167, 176, 179, 185, 188, 191, 257, 260, 263, 272, 275, 281, 284, 287, 320, 323, 329, 332, 335, 353, 356, 359, 368, 371, 377, 380, 383, 512, 515, 521, 524, 527, 545, 548, 551, 560, 563, 569, 572, 575

Offset: 1

Author

Antti Karttunen, Oct 01 2023

Keywords

• nonn

Comments

The sequence is defined inductively as:

(a) it contains 0 and 2,

and

(b) for any nonzero term a(n), (2*a(n)) + 1 and 4*a(n) are also included as terms.

Because the inductive definition guarantees that all terms after 0 are of the form 3k+2 (A016789), and because for any n >= 0, n^2 == 0 or 1 (mod 3), (i.e., squares are in A032766), it follows that there are no squares in this sequence after the initial 0.

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000290, A010052, A032766, A163511, A365807 (characteristic function).

Positions of even terms in A365805.

Sequence A243071(n^2), n >= 1, sorted into ascending order.

Subsequences: A004171, A055010, A365809 (odd terms).

Subsequence of A016789 (after the initial 0).

Cf. also A277335, A365801, A365802, A365806.

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));
isA365808v2(n) = issquare(A163511(n));

PARI

isA365808(n) = if(n<=2, !(n%2), if(n%2, isA365808((n-1)/2), if(n%4, 0, isA365808(n/4))));

Python

from itertools import count, islice
def A365808_gen(): # generator of terms
    return map(lambda n:(3*(n+1)>>2)-1,filter(lambda n:n==1 or (n&3==3 and not '00' in bin(n)),count(1)))
A365808_list = list(islice(A365808_gen(),20)) # Chai Wah Wu, Feb 12 2025