A064372 Additive function a(n) defined by the recursive formula a(1)=1 and a(p^k)=a(k) for any prime p.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

Offset

1,6

Comments

That is, if i, j, k, ... are relatively prime, then a(i*j*k*...) = a(i) + a(j) + a(k) + ... - N. J. A. Sloane, Nov 20 2007

Starts almost the same as A001221 (the number of distinct primes dividing n): the first twelve terms which are different are a(1), a(64), a(192), a(320), a(448), a(576), a(704), a(729), a(832), a(960), a(1024) and a(1088), since the first non-unitary values of n are a(6) and(10). - Henry Bottomley, Sep 23 2002

a(A164336(n)) = 1. - Reinhard Zumkeller, Aug 27 2011

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = A106491(n) - A106490(n) = A106495(A106444(n)). - Antti Karttunen, May 09 2005

a(1) = 1, a(n) = Sum_{k=1..A001221(n)} a(A124010(n,k)) for n > 1. - Reinhard Zumkeller, Aug 27 2011

Example

a(30) = a(5^1 * 3^1 * 2^1) = a(1) + a(1) + a(1) = 3.

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      add(a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Aug 23 2020

Mathematica

a[1] = 1; a[n_] := a[n] = Plus @@ a /@ FactorInteger[n][[All, 2]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Sep 19 2012 *)

Prog

(Haskell)
a064372 1 = 1
a064372 n = sum $ map a064372 $ a124010_row n
-- Reinhard Zumkeller, Aug 27 2011

Crossrefs

Cf. A001221, A079553, A106444, A106490, A106491, A106495, A124010, A164336.

Sequence in context: A087802 A079553 A001221 * A343943 A370817 A345926

Adjacent sequences: A064369 A064370 A064371 * A064373 A064374 A064375

Keyword

nonn,easy,nice

Author

Steven Finch, Sep 26 2001

Status

approved