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

1,3

Comments

a(n)<n for n>1.

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

Links

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

Formula

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

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

Example

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.

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 *)

Prog

(SageMath) A072084 = lambda n: prod(p.digits(base=2).count(1)**m for p, m in factor(n)) # D. S. McNeil, Jan 17 2011
(Haskell)
a072084 = product . map a000120 . a027746_row
-- Reinhard Zumkeller, Feb 10 2013
(PARI) a(n)=my(f=factor(n)); f[, 1]=apply(hammingweight, f[, 1]); factorback(f) \\ Charles R Greathouse IV, Aug 06 2015

Crossrefs

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

Sequence in context: A355731 A353394 A349494 * A359587 A336471 A357975

Adjacent sequences: A072081 A072082 A072083 * A072085 A072086 A072087

Keyword

nonn,base,mult

Author

Reinhard Zumkeller, Jun 14 2002

Status

approved