A072084 In prime factorization of n replace all primes with their numbers of 1's in binary representation.

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

Offset: 1

Author

Reinhard Zumkeller, Jun 14 2002

Keywords

• nonn
• base
• mult

Comments

a(n)1.

Multiplicative with a(p^e) = A000120(p)^e. - Mitch Harris, Apr 19 2005

Examples

			a(30) = a(2*3*5) = a(2)*a(3)*a(5) = 1*2*2 = 4,
as a(2)=a('10')=1, a(3)=a('11')= 2 and a(5)=a('101')=2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000120, A072085, A072087, A027746, A014499.

Programs

Haskell

a072084 = product . map a000120 . a027746_row
-- Reinhard Zumkeller, Feb 10 2013

Maple

A072084 := proc(n) local a,c; readlib(ifactors):
a := n -> add(i,i=convert(n, base, 2));
mul(a(c[1])^c[2],c=ifactors(n)[2]) end:
# Peter Luschny, Jan 16 2010

Mathematica

a[n_] := Times @@ Power @@@ (FactorInteger[n] /. {p_Integer, e_} :> {DigitCount[p, 2, 1], e}); Array[a, 100] (* Jean-François Alcover, Feb 09 2018 *)

PARI

a(n)=my(f=factor(n)); f[,1]=apply(hammingweight, f[,1]); factorback(f) \\ Charles R Greathouse IV, Aug 06 2015

Sage

A072084 = lambda n: prod(p.digits(base=2).count(1)**m for p,m in factor(n)) # D. S. McNeil, Jan 17 2011

Formula

Completely multiplicative with a(p) = number of 1's in binary representation of prime p.

Multiplicative with a(p^e) = A000120(p)^e