A252463 - OEIS

%I #33 Jan 21 2023 03:27:17

%S 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,

%T 15,29,16,14,17,15,18,31,19,22,20,37,21,41,22,12,23,43,24,25,25,26,26,

%U 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

%N 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.

%C 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).

%H Antti Karttunen, <a href="/A252463/b252463.txt">Table of n, a(n) for n = 1..8192</a>

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

%F Other identities. For all n >= 1:

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

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

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

%F 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

%t Table[Which[n == 1, 1, EvenQ@ n, n/2, True, Times @@ Power[

%t Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n], {n, 81}] (* _Michael De Vlieger_, Sep 16 2017 *)

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

%o (Python)

%o from sympy import factorint, prevprime

%o from operator import mul

%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 a(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)

%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Sep 15 2017

%o (PARI) a064989(n) = factorback(Mat(apply(t->[max(precprime(t[1]-1), 1), t[2]], Vec(factor(n)~))~)); \\ A064989

%o a(n) = if (n==1, 1, if (n%2, a064989(n), n/2)); \\ _Michel Marcus_, Oct 13 2021

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

%Y Bisections: A000027 and A064216.

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

%K nonn,look

%O 1,3

%A _Antti Karttunen_, Dec 20 2014