A292245 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+1 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

1, 2, 2, 5, 4, 4, 11, 4, 8, 17, 10, 18, 9, 8, 22, 17, 8, 8, 17, 22, 36, 41, 8, 42, 17, 16, 44, 21, 34, 32, 35, 20, 32, 33, 36, 64, 69, 18, 34, 73, 16, 74, 37, 44, 82, 33, 34, 34, 89, 16, 64, 69, 16, 68, 65, 34, 64, 33, 44, 64, 33, 72, 16, 65, 82, 68, 85, 16, 128, 137, 84, 72, 69, 34, 138, 145, 32, 84, 145, 88, 88, 149, 42, 162, 65, 68, 164, 45, 64

Offset: 1

Author

Antti Karttunen, Sep 15 2017

Keywords

• nonn
• base

Examples

			For n=1 (the termination value of the iteration), 1 is of the form 3k+1, thus a(1) = 1*(2^0) = 1.
For n=2, 2 is not of the form 3k+1, while A253889(2) = 1 is, thus a(2) = 0*(2^0) + 1*2(^1) = 2.
For n=4, 4 is of the form 3k+1, while A253889(4) = 2 is not, but then A253889(2) = 1 again is, thus a(4) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A048673, A064216, A289813, A292243, A292244, A292246, A292248.

Programs

Mathematica

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Map[FromDigits[#, 2] &[IntegerDigits[#, 3] /. 2 -> 0] &, Array[a, 98]] (* Michael De Vlieger, Sep 16 2017 *)

Formula

a(1) = 1; for n > 1, a(n) = 2*a(A253889(n)) + [n ≡ 1 (mod 3)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 3k+1, and 0 otherwise.

a(n) = A289813(A292243(n)).

Other identities. For all n >= 1:

a(A048673(n)) = A292248(n).

a(n) + A292244(n) = A064216(n).

a(n) AND A292244(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).