A292383 - OEIS

%I #22 Sep 21 2017 21:21:41

%S 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,

%T 358,10,717,0,20,88,11,0,1434,178,45,16,2868,16,5737,44,8,358,11475,

%U 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

%N 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.

%H Antti Karttunen, <a href="/A292383/b292383.txt">Table of n, a(n) for n = 1..2048</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F 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.

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

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

%F Other identities. For n >= 1:

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

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

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

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

%e 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.

%e 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.

%e 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.

%t 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 *)

%o (PARI)

%o 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)};

%o A252463(n) = if(!(n%2),n/2,A064989(n));

%o A292383(n) = if(1==n,0,(if(3==(n%4),1,0)+(2*A292383(A252463(n)))));

%o (Scheme) (define (A292383 n) (A292373 (A292384 n)))

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

%K nonn,base

%O 1,5

%A _Antti Karttunen_, Sep 15 2017