A252463 Hybrid shift: a(1) = 1, a(2n) = n, a(2n+1) = A064989(2n+1); shift the even numbers one bit right, shift the prime factorization of odd numbers one step towards smaller primes.

1, 1, 2, 2, 3, 3, 5, 4, 4, 5, 7, 6, 11, 7, 6, 8, 13, 9, 17, 10, 10, 11, 19, 12, 9, 13, 8, 14, 23, 15, 29, 16, 14, 17, 15, 18, 31, 19, 22, 20, 37, 21, 41, 22, 12, 23, 43, 24, 25, 25, 26, 26, 47, 27, 21, 28, 34, 29, 53, 30, 59, 31, 20, 32, 33, 33, 61, 34, 38, 35, 67, 36, 71, 37, 18, 38, 35, 39, 73, 40, 16

Offset: 1

Author

Antti Karttunen, Dec 20 2014

Keywords

• nonn
• look

Comments

For any node n >= 2 in binary trees A005940 and A163511, a(n) gives the parent node of n. (Here we assume that their initial root 1 is its own parent).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192

Crossrefs

A252464 gives the number of iterations needed to reach 1 from n.

Bisections: A000027 and A064216.

Cf. A000035, A001222, A004526, A005940, A064989, A151799, A163511, A246277, A252461, A252462.

Programs

Mathematica

Table[Which[n == 1, 1, EvenQ@ n, n/2, True, Times @@ Power[
Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n], {n, 81}] (* Michael De Vlieger, Sep 16 2017 *)

PARI

a064989(n) = factorback(Mat(apply(t->[max(precprime(t[1]-1), 1), t[2]], Vec(factor(n)~))~)); \\ A064989
a(n) = if (n==1, 1, if (n%2, a064989(n), n/2)); \\ Michel Marcus, Oct 13 2021

Python

from sympy import factorint, prevprime
from operator import mul
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 a(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Sep 15 2017

Scheme

(define (A252463 n) (cond ((<= n 1) n) ((even? n) (/ n 2)) (else (A064989 n))))

Formula

a(1) = 1, a(2n) = n, a(2n+1) = A064989(2n+1).

Other identities. For all n >= 1:

a(2n-1) = A064216(n).

A001222(a(n)) = A001222(n) - (1 - A000035(n)).

Above means: if n is odd, A001222(a(n)) = A001222(n) and if n is even, A001222(a(n)) = A001222(n) - 1.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/8 + (1/2) * Product_{p prime > 2} ((p^2-p)/(p^2-q(p))) = 0.2905279467..., where q(p) = prevprime(p) (A151799). - Amiram Eldar, Jan 21 2023