A324183 - OEIS

%I #12 Jul 25 2023 21:04:37

%S 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,

%T 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,

%U 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

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

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

%H Antti Karttunen, <a href="/A324183/b324183.txt">Table of n, a(n) for n = 0..65537</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

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

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

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

%o (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)));

%o (PARI)

%o 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));

%o A324183(n) = numdiv(A163511(n));

%o (PARI)

%o A054429(n) = if(!n,n,((3<<#binary(n\2))-n-1)); \\ After code in A054429

%o A106737(n) = sum(k=0, n, (binomial(n+k, n-k)*binomial(n, k)) % 2);

%o A324183(n) = A106737(A054429(n));

%o (Python)

%o def A324183(n):

%o     if n:

%o         c = 1

%o         while n:

%o             c *= (s:=(~n&n-1).bit_length()+1)

%o             n >>= s

%o         return c*(s+1)//s

%o     return 1 # _Chai Wah Wu_, Jul 25 2023

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

%K nonn

%O 0,2

%A _Antti Karttunen_, Feb 17 2019