A292383 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

0, 0, 1, 0, 2, 2, 5, 0, 0, 4, 11, 4, 22, 10, 5, 0, 44, 0, 89, 8, 8, 22, 179, 8, 0, 44, 1, 20, 358, 10, 717, 0, 20, 88, 11, 0, 1434, 178, 45, 16, 2868, 16, 5737, 44, 8, 358, 11475, 16, 0, 0, 89, 88, 22950, 2, 17, 40, 176, 716, 45901, 20, 91802, 1434, 17, 0, 40, 40, 183605, 176, 356, 22, 367211, 0, 734422, 2868, 1, 356, 22, 90, 1468845, 32, 0, 5736, 2937691, 32

Offset

1,5

Links

Antti Karttunen, Table of n, a(n) for n = 1..2048

Index entries for sequences related to binary expansion of n

Index entries for sequences computed from indices in prime factorization

Formula

a(1) = 0; for n > 1, a(n) = 2*a(A252463(n)) + [n ≡ 3 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+3, and 0 otherwise.

a(n) = A292373(A292384(n)).

a(n) = A292274(A243071(n)).

Other identities. For n >= 1:

a(2n) = 2*a(n).

a(n) + A292385(n) = A243071(n).

a(A163511(n)) = A292274(n).

A000120(a(n)) = A292377(n).

Example

For n = 3, the starting value is of the form 4k+3, after which follows A252463(3) = 2, and A252463(2) = 1, the end point of iteration, and neither 2 nor 1 is of the form 4k+3, thus a(3) = 1*(2^0) + 0*(2^1) + 0*(2^2) = 1.
For n = 5, the starting value is not of the form 4k+3, after which follows A252463(5) = 3 (which is), continuing as before as 3 -> 2 -> 1, thus a(5) = 0*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 2.
For n = 10, the starting value is not of the form 4k+3, after which follows A252463(10) = 5 (also not 4k+3), and then A252463(5) = 3 (which is), continuing as before as 3 -> 2 -> 1, thus a(10) = 0*(2^0) + + 0*(2^1) + 1*(2^2) + 0*(2^3) + 0*(2^4) = 4.

Mathematica

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 3, 1, 0], 2], {n, 84}] (* Michael De Vlieger, Sep 21 2017 *)

Prog

(PARI)
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
A252463(n) = if(!(n%2), n/2, A064989(n));
A292383(n) = if(1==n, 0, (if(3==(n%4), 1, 0)+(2*A292383(A252463(n)))));
(Scheme) (define (A292383 n) (A292373 (A292384 n)))

Crossrefs

Cf. A163511, A243071, A292274, A292373, A292377, A292380, A292381, A292382, A292384, A292385.

Sequence in context: A012858 A366756 A363842 * A332896 A100247 A194123

Adjacent sequences: A292380 A292381 A292382 * A292384 A292385 A292386

Keyword

nonn,base

Author

Antti Karttunen, Sep 15 2017

Status

approved