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A072084 In prime factorization of n replace all primes with their numbers of 1's in binary representation. 7
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 09 2018 *)

PROG

(Sage) 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: A295283 A103284 A071287 * A133755 A070104 A131085

Adjacent sequences:  A072081 A072082 A072083 * A072085 A072086 A072087

KEYWORD

nonn,base,mult

AUTHOR

Reinhard Zumkeller, Jun 14 2002

STATUS

approved

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Last modified November 12 04:21 EST 2019. Contains 329051 sequences. (Running on oeis4.)