A353412 The odd part of hybrid shift: a(n) = A000265(A252463(n)).

1, 1, 1, 1, 3, 3, 5, 1, 1, 5, 7, 3, 11, 7, 3, 1, 13, 9, 17, 5, 5, 11, 19, 3, 9, 13, 1, 7, 23, 15, 29, 1, 7, 17, 15, 9, 31, 19, 11, 5, 37, 21, 41, 11, 3, 23, 43, 3, 25, 25, 13, 13, 47, 27, 21, 7, 17, 29, 53, 15, 59, 31, 5, 1, 33, 33, 61, 17, 19, 35, 67, 9, 71, 37, 9, 19, 35, 39, 73, 5, 1, 41, 79, 21, 39, 43, 23, 11

Offset: 1

Author

Antti Karttunen, Apr 18 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000005, A252463, A064216, A064989, A320107.

Cf. A000265 (even bisection), A353413 (odd bisection).

Programs

PARI

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A252463(n) = if(!(n%2),n/2,A064989(n));
A353412(n) = A000265(A252463(n));

Python

from math import prod
from sympy import factorint, prevprime
def A353412(n): return int(bin(prod(1 if p == 2 else prevprime(p)*e for p, e in factorint(n).items()) if n % 2 else n//2)[2:].rstrip('0'),2) # Chai Wah Wu, Apr 18 2022

Formula

a(n) = A000265(A252463(n)).

a(2*n) = A000265(n), a(2*n-1) = A353413(n) = A000265(A064216(n)).

For all n >= 1, A000005(a(n)) = A320107(n).