A324183 a(n) = d(A163511(n)), where d(n) is A000005, the number of divisors of n.

1, 2, 3, 2, 4, 3, 4, 2, 5, 4, 6, 3, 6, 4, 4, 2, 6, 5, 8, 4, 9, 6, 6, 3, 8, 6, 8, 4, 6, 4, 4, 2, 7, 6, 10, 5, 12, 8, 8, 4, 12, 9, 12, 6, 9, 6, 6, 3, 10, 8, 12, 6, 12, 8, 8, 4, 8, 6, 8, 4, 6, 4, 4, 2, 8, 7, 12, 6, 15, 10, 10, 5, 16, 12, 16, 8, 12, 8, 8, 4, 15, 12, 18, 9, 18, 12, 12, 6, 12, 9, 12, 6, 9, 6, 6, 3, 12, 10, 16, 8, 18, 12, 12, 6, 16, 12

Offset: 0

Author

Antti Karttunen, Feb 17 2019

Keywords

• nonn

Comments

For all i, j: A286531(i) = A286531(j) => a(i) = a(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000005, A054429, A106737, A163511, A286531, A324184, A324187.

Programs

PARI

A324183(n) = if(!n,1,n = ((3<<#binary(n\2))-n-1); my(e=0,m=1); while(n>0, if(!(n%2), m *= (1+e); e=0, e++); n >>= 1); (m*(1+e)));

PARI

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A324183(n) = numdiv(A163511(n));

PARI

A054429(n) = if(!n,n,((3<<#binary(n\2))-n-1)); \\ After code in A054429
A106737(n) = sum(k=0, n, (binomial(n+k, n-k)*binomial(n, k)) % 2);
A324183(n) = A106737(A054429(n));

Python

def A324183(n):
    if n:
        c = 1
        while n:
            c *= (s:=(~n&n-1).bit_length()+1)
            n >>= s
        return c*(s+1)//s
    return 1 # Chai Wah Wu, Jul 25 2023

Formula

a(n) = A000005(A163511(n)).

a(n) = A106737(A054429(n)).

For all n >= 0, a(2^n) = n+2.