A338692 Positions of 1's in A209615.

1, 4, 5, 6, 9, 13, 14, 16, 17, 20, 21, 22, 24, 25, 29, 30, 33, 36, 37, 38, 41, 45, 46, 49, 52, 53, 54, 56, 57, 61, 62, 64, 65, 68, 69, 70, 73, 77, 78, 80, 81, 84, 85, 86, 88, 89, 93, 94, 96, 97, 100, 101, 102, 105, 109, 110, 113, 116, 117, 118, 120, 121, 125, 126

Offset

1,2

Comments

Also positions of 1's and 4's in A003324.

Also positions of 0's in A292077. - Jianing Song, Nov 27 2021

Numbers of the form (2*k+1) * 2^e where k+e is even. In other words, union of {(4*m+1) * 2^(2t)} and {(4*m+3) * 2^(2t+1)}, where m >= 0, t >= 0.

Numbers whose quaternary (base-4) expansion ends in 100...00 or 1200..00 or 3200..00. Trailing 0's are not necessary.

There are precisely 2^(N-1) terms <= 2^N for every N >= 1.

Equals A016813 Union A343501.

Complement of A338691. - Jianing Song, Apr 26 2021

Links

Jianing Song, Table of n, a(n) for n = 1..8192

Kevin Ryde, Iterations of the Alternate Paperfolding Curve, see index "TurnLeft" with a(n) = TurnLeft(n-1).

Formula

a(n) = A343500(n)/2. - Jianing Song, Apr 26 2021

Example

14 is a term since it is in the family {(4*m+3) * 2^(2t+1)} with m = 1, t = 0.
16 is a term since it is in the family {(4*m+1) * 2^(2t)} with m = 0, t = 2.

Mathematica

A338692Q[k_] := JacobiSymbol[-1, k]*(-1)^IntegerExponent[k, 2] == 1;
Select[Range[200], A338692Q] (* Paolo Xausa, Feb 26 2025 *)

Prog

(PARI) isA338692(n) = my(e=valuation(n, 2), k=bittest(n, e+1)); !((k+e)%2)
(Python)
def A338692(n):
    def f(x): return n+sum(((x>>i)-1>>2)+1 for i in range(1, x.bit_length(), 2))+sum(((x>>i)-3>>2)+1 for i in range(0, x.bit_length(), 2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 27 2025

Crossrefs

Cf. A209615, A338691 (positions of (-1)'s), A016813 (the odd terms), A343501 (the even terms), A003324, A292077, A343500.

Sequence in context: A245236 A209122 A073263 * A039013 A241511 A020669

Adjacent sequences: A338689 A338690 A338691 * A338693 A338694 A338695

Keyword

nonn,easy

Author

Jianing Song, Apr 24 2021

Status

approved