A292382 Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+2 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

0, 1, 2, 2, 4, 5, 8, 4, 4, 9, 16, 10, 32, 17, 10, 8, 64, 9, 128, 18, 18, 33, 256, 20, 8, 65, 8, 34, 512, 21, 1024, 16, 34, 129, 20, 18, 2048, 257, 66, 36, 4096, 37, 8192, 66, 20, 513, 16384, 40, 16, 17, 130, 130, 32768, 17, 36, 68, 258, 1025, 65536, 42, 131072, 2049, 36, 32, 68, 69, 262144, 258, 514, 41, 524288, 36, 1048576, 4097, 18, 514, 40

Offset: 1

Author

Antti Karttunen, Sep 15 2017

Keywords

• nonn
• base

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2048
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A005940, A156552, A292272, A292372, A292380, A292381, A292383, A292384.

Programs

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] == 2, 1, 0], 2], {n, 77}] (* Michael De Vlieger, Sep 21 2017 *)

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));
A292382(n) = if(1==n,0,(if(2==(n%4),1,0)+(2*A292382(A252463(n)))));

PARI

a(n) = my(m=factor(n),k=-2); sum(i=1,matsize(m)[1], 1 << (primepi(m[i,1]) + (k+=m[i,2]))); \\ Kevin Ryde, Dec 11 2020

Python

from sympy.core.cache import cacheit
from sympy.ntheory.factor_ import digits
from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a292372(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join(['1' if i==2 else '0' for i in k]), 2)
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4
def a(n): return a292372(a292384(n))
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

Scheme

(define (A292382 n) (A292372 (A292384 n)))

Formula

a(n) = A292272(A156552(n)).

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

a(n) = A292372(A292384(n)).

Other identities. For n >= 1:

a(n) AND A292380(n) = 0, where AND is a bitwise-AND (A004198).

a(n) + A292380(n) = A156552(n).

A000120(a(n)) + A000120(A292380(n)) = A001222(n).