A322862 a(n) = A000120(A285330(n)).

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 4, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 4, 1, 2, 3, 1, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 4, 3, 2, 3, 1, 3, 4, 2, 1, 3, 2, 2, 2, 3, 1, 4, 2, 4, 2, 2, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 3

Offset: 1

Author

Antti Karttunen, Dec 31 2018

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000120, A001221, A001222, A285328, A285330, A322811, A322861, A322869.

Programs

Mathematica

Table[DigitCount[#, 2, 1] &@ Which[n == 1, 0, MoebiusMu@ n != 0, Total@ Map[#2*2^(PrimePi@ #1 - 1) & @@ # &, FactorInteger[n]], True, With[{r = DivisorSum[n, EulerPhi[#] Abs@ MoebiusMu[#] &]}, SelectFirst[Range[n - 2, 2, -1], DivisorSum[#, EulerPhi[#] Abs@ MoebiusMu[#] &] == r &]]], {n, 105}] (* Michael De Vlieger, Dec 31 2018 *)

PARI

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(moebius(n)<>0,A048675(n),A285328(n));
A322862(n) = hammingweight(A285330(n));
\\ Or just as:
A322862(n) = if(issquarefree(n), omega(n), hammingweight(A285328(n)));

Formula

a(n) = A000120(A285330(n)).

If n is squarefree, a(n) = A322869(n) = A000120(A048675(n)) = A001221(n), otherwise a(n) = A000120(A285328(n)).