A353412 - OEIS

%I #18 Apr 18 2022 22:33:42

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A353412/b353412.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

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

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

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

%o (PARI)

%o A000265(n) = (n>>valuation(n,2));

%o A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };

%o A252463(n) = if(!(n%2),n/2,A064989(n));

%o A353412(n) = A000265(A252463(n));

%o (Python)

%o from math import prod

%o from sympy import factorint, prevprime

%o 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

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

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

%K nonn

%O 1,5

%A _Antti Karttunen_, Apr 18 2022