A048716 Numbers k such that (k AND 4*k) = 0, where AND is bitwise-and.

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201, 204, 256, 257, 258, 259, 260, 262, 264, 265

Offset

1,3

Comments

"Numbers k such that binary expansion matches ((0)*00(1?)1)*(0*)." - This was the original, somewhat obscure definition involving regexp-syntax.

If n is a term, then 2*n is also a term, and vice versa.

If bit i is 1, then bits i+-2 must be 0.

Numbers k such that 5*k = A048725(k) = A048720(5, k), where A048720 is carryless base-2 multiplication. (Edited by Antti Karttunen, Dec 21 2025)

It appears that k is in the sequence if and only if C(5k,k) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003

Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011

A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Wikipedia, Regular expression

Index entries for 2-automatic sequences.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Index entries for sequences defined by congruent products under XOR

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to carryless arithmetic.

Formula

{k such that A004198(k, 4*k) = 0}. - Antti Karttunen, Dec 21 2025

Mathematica

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Alternative: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{___, 0, _, 1, ___}|{___ 1, _, 0, ___}] && !MatchQ[bb, {___, 1, _, 1, ___}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

Prog

(PARI) is(n)=!bitand(n, n>>2) \\ Charles R Greathouse IV, Oct 03 2016
(PARI) list(lim)=my(v=List(), n, t); while(n<=lim, t=bitand(n, n>>2); if(t, n+=1<<valuation(t, 2), listput(v, n); n++)); Vec(v) \\ Charles R Greathouse IV, Oct 22 2021

Crossrefs

Intersection of A004742 and A003726.

Subsequence of A391740.

Subsequences: A048715, A048719, A115422.

Positions of 0's in A048729.

Cf. A003714, A004198, A048720, A048725, A115845, A115847, A116360.

Sequence in context: A019469 A081491 A386917 * A010434 A074230 A064438

Adjacent sequences: A048713 A048714 A048715 * A048717 A048718 A048719

Keyword

nonn,base,easy

Author

Antti Karttunen, Mar 30 1999

Extensions

Name changed and more terms added, and an erroneous claim in the Crossrefs-section corrected by Antti Karttunen, Dec 21 2025

Status

approved