A365809 -id:A365809 - cp's OEIS frontend

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

Antti Karttunen, Oct 01 2023

nonn

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.

Index entries for sequences related to binary expansion of n

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.

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));

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

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

A380358 Numbers whose binary expansion ends with 11 and does not contain adjacent zeros.

Original entry on oeis.org

3, 7, 11, 15, 23, 27, 31, 43, 47, 55, 59, 63, 87, 91, 95, 107, 111, 119, 123, 127, 171, 175, 183, 187, 191, 215, 219, 223, 235, 239, 247, 251, 255, 343, 347, 351, 363, 367, 375, 379, 383, 427, 431, 439, 443, 447, 471, 475, 479, 491, 495, 503, 507, 511, 683

Offset: 1

R. J. Cintra, Jan 22 2025

The numbers in this sequence appear in the conversion of conventional binary numbers to the canonical signed-digit representation.

			183 is in the sequence because its binary expansion is 10110111.

J. L. Smith and A. Weinberger, "Shortcut Multiplication for Binary Digital Computers", in Methods for High-Speed Addition and Multiplication, National Bureau of Standards Circular 591, Sec. 1, February, 1958, page 21.

R. J. Cintra, A note on the conversion of nonnegative integers to the canonical signed-digit representation, arXiv:2501.10908 [eess.SP], 2025.

Cf. A003754, A052499, A247648, A365808, A365809.

Select[4*Range[0, 170] + 3, SequencePosition[IntegerDigits[#, 2], {0, 0}] == {} &] (* Amiram Eldar, Feb 05 2025 *)

from itertools import count, islice
def A380358_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n&3==3 and not '00' in bin(n),count(max(startvalue,1)))
A380358_list = list(islice(A380358_gen(),20)) # Chai Wah Wu, Feb 12 2025

a(n) = 2 * A247648(n) + 1.
From Hugo Pfoertner, Feb 07 2025: (Start)
a(n) = 4*A052499(n) - 1.
a(n) = 4*(A365808(n+1) + 1)/3 - 1.
a(n) = 2*(A365809(n) + 1)/3 - 1. (End)

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

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A380358 Numbers whose binary expansion ends with 11 and does not contain adjacent zeros.

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