A292381 - OEIS

%I #23 May 15 2021 06:18:13

%S 1,2,4,4,9,8,18,8,9,18,36,16,73,36,16,16,147,18,294,36,37,72,588,32,

%T 19,146,16,72,1177,32,2354,32,73,294,32,36,4709,588,144,72,9419,74,

%U 18838,144,33,1176,37676,64,39,38,292,292,75353,32,74,144,589,2354,150706,64,301413,4708,72,64,147,146,602826,588,1177,64,1205652,72,2411305,9418,36,1176

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

%H Antti Karttunen, <a href="/A292381/b292381.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) = 1; for n > 1, a(n) = 2*a(A252463(n)) + [n ≡ 1 (mod 4)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 4k+1, and 0 otherwise.

%F a(n) = A292371(A292384(n)).

%F Other identities. For n >= 1:

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

%F A000120(a(n)) = A292375(n).

%F For n >= 2, a(n) = A004754(A292385(n)).

%e For n = 1, the starting value (which is also the ending point) is of the form 4k+1, thus a(1) = 1*(2^0) = 1.

%e For n = 2, the starting value is not of the form 4k+1, but its parent, A252463(2) = 1 is, thus a(2) = 0*(2^0) + 1*(2^1) = 2.

%e For n = 3, the starting value is not of the form 4k+1, after which follows 2 (also not 4k+1), and then 2 -> 1, and it is only the end-point of iteration which is of the form 4k+1, thus a(3) = 0*(2^0) + 0*(2^1) + 1*(2^2) = 4.

%e For n = 5, the starting value is of the form 4k+1, after which follows A252463(5) = 3 (which is not), and then continuing as before as 3 -> 2 -> 1, thus a(5) = 1*(2^0) + 0*(2^1) + 0*(2^2) + 1*(2^3) = 9.

%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] == 1, 1, 0], 2], {n, 76}] (* _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 A292381(n) = if(1==n,n,(if(1==(n%4),1,0)+(2*A292381(A252463(n)))));

%o (Scheme) (define (A292381 n) (A292371 (A292384 n)))

%o (Python)

%o from sympy.core.cache import cacheit

%o from sympy.ntheory.factor_ import digits

%o from sympy import factorint, prevprime

%o from operator import mul

%o from functools import reduce

%o def a292371(n):

%o     k=digits(n, 4)[1:]

%o     return 0 if n==0 else int("".join(['1' if i==1 else '0' for i in k]), 2)

%o def a064989(n):

%o     f=factorint(n)

%o     return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])

%o def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)

%o @cacheit

%o def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4

%o def a(n): return a292371(a292384(n))

%o print([a(n) for n in range(1, 111)]) # _Indranil Ghosh_, Sep 21 2017

%Y Cf. A252463, A292371, A292375, A292380, A292382, A292383, A292384, A292385.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Sep 15 2017