A343501 Positions of 4's in A003324.

4, 6, 14, 16, 20, 22, 24, 30, 36, 38, 46, 52, 54, 56, 62, 64, 68, 70, 78, 80, 84, 86, 88, 94, 96, 100, 102, 110, 116, 118, 120, 126, 132, 134, 142, 144, 148, 150, 152, 158, 164, 166, 174, 180, 182, 184, 190, 196, 198, 206, 208, 212, 214, 216, 222, 224, 228

Offset

1,1

Comments

Numbers of the form (2*k+1) * 2^e where e >= 1, 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. At least one trailing zero is required in the first case but not in the latter two cases.

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

Also even indices of 1 in A209615. - Jianing Song, Apr 24 2021

Complement of A343500 with respect to the even numbers. - Jianing Song, Apr 26 2021

Links

Jianing Song, Table of n, a(n) for n = 1..16384 (all terms <= 2^16).

Formula

a(n) = 2*A338691(n). - Hugo Pfoertner, Apr 26 2021

Example

A003324 starts with 1, 2, 3, 4, 1, 4, 3, 2, 1, 2, 3, 2, 1, 4, 3, 4, ... We have A003324(4) = A003324(6) = A003324(14) = A003324(16) = ... = 4, so this sequence starts with 4, 6, 14, 16, ...

Mathematica

okQ[n_] := If[OddQ[n], False, Module[{e = IntegerExponent[n, 2], k}, k = (n/2^e - 1)/2; EvenQ[k + e]]];
Select[Range[1000], okQ] (* Jean-François Alcover, Apr 19 2021, after PARI *)

Prog

(PARI) isA343501(n) = if(n%2, 0, my(e=valuation(n, 2), k=bittest(n, e+1)); !((k+e)%2))

Crossrefs

Cf. A003324, A343500 (positions of 2's), A209615, A338691.

Even terms in A338692.

Sequence in context: A102029 A310618 A310619 * A029641 A089377 A310620

Adjacent sequences: A343498 A343499 A343500 * A343502 A343503 A343504

Keyword

nonn,easy

Author

Jianing Song, Apr 17 2021

Status

approved