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

1,3

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

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

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 *)

Prog

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

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.

Sequence in context: A214288 A332825 A253554 * A332893 A113605 A369984

Adjacent sequences: A252460 A252461 A252462 * A252464 A252465 A252466

Keyword

nonn,look

Author

Antti Karttunen, Dec 20 2014

Status

approved