A281681 a(n) = A055396(A071904(n)) - 1.

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 4, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 2, 1, 3, 1, 5, 1, 2, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 4, 1, 2, 3, 1, 5, 1, 1, 2, 1, 1, 2, 5, 1, 4, 1, 3, 1, 2, 1, 1, 2, 1, 1, 3, 6, 1, 2, 1, 5, 3, 1, 2, 1, 1, 4, 1, 6, 2, 1, 3, 1, 2, 1, 4, 3, 1, 1, 2, 1, 7, 1, 2, 1, 3, 1, 5, 1, 2, 1, 6, 1, 2, 1, 5, 1, 4, 1, 3, 2, 1

Offset: 1

Author

Enrique Navarrete, Jan 26 2017

Keywords

• nonn

Comments

The sequence measures, in a sense, inversions in remainders of odd numbers upon factoring out their largest divisors (see A281680).

In A281680, we have A281680(4) = A281680(7) = A281680(10) = 3 (and there will be infinitely many 1's to the right after each one of them), so there is why a(1)=a(2)=a(3)=1. Then we have A281680(12) = 5 (and there will be infinitely many 1's and 3's to the right), so that's why a(4) = 2, and so forth. I used 1,2,3,... here to represent these inversions, but any other symbols could have been used.

Entries correspond to the position of the lowest prime factor of the odd composites, with prime=3 being position 1. - Bill McEachen, Jan 28 2018

Links

• Bill McEachen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A055396, A071904, A281680, A162022.

Programs

PARI

genit(maxx)={forcomposite(i5=9,maxx,if(i5%2==0,next);ptr=0;forprime(x=3,maxx,ptr+=1;if(i5%x==0,print1(ptr,",");break)));} \\ Bill McEachen, Jan 28 2018

Python

from sympy import primepi, primefactors
def A281681(n):
    if n == 1: return 1
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return primepi(min(primefactors(m)))-1 # Chai Wah Wu, Aug 02 2024

Extensions

Name changed by Robert Israel, Aug 03 2020