A064553 a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.

1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16, 8, 18, 9, 16, 15, 12, 10, 24, 16, 14, 27, 20, 11, 24, 12, 32, 18, 16, 20, 36, 13, 18, 21, 32, 14, 30, 15, 24, 36, 20, 16, 48, 25, 32, 24, 28, 17, 54, 24, 40, 27, 22, 18, 48, 19, 24, 45, 64, 28, 36, 20, 32, 30, 40, 21, 72, 22, 26

Offset: 1

Author

Reinhard Zumkeller, Sep 21 2001

Keywords

• mult
• nice
• nonn
• look

Comments

a(n) <= n for all n and a(x) = x iff x = 2^i * 3^j for i, j >= 0: a(A003586(n)) = A003586(n) for n > 0. By definition a is completely multiplicative and also surjective. a(p) < a(q) for primes p < q.

Completely multiplicative with a(prime(i)) = i + 1. - Charles R Greathouse IV, Sep 07 2012

a(A080688(n,k)) = A080444(n,k) = n for k=1..A001055(n). - Reinhard Zumkeller, Oct 01 2012

Examples

			a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..8000
• T. D. Noe, Plot of A064553
• Index to divisibility sequences
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000005, A000040, A003961, A003963, A049084, A020639, A064554, A064555, A001055, A003586, A064557, A064558, A027746, A027748, A124010, A181819.

A left inverse of A290641.

Programs

Haskell

a064553 1 = 1
a064553 n = product $ map ((+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Feb 17 2012, Jan 28 2011

Maple

A064553 := proc(n)
    local a,f,p,e ;
    a := 1 ;
    for f in ifactors(n)[2] do
        p :=op(1,f) ;
        e :=op(2,f) ;
        a := a*(numtheory[pi](p)+1)^e ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 07 2012

Mathematica

nn=100; a=Table[0, {nn}]; a[[1]]=1; Do[If[PrimeQ[i], a[[i]]=PrimePi[i]+1, p=FactorInteger[i][[1,1]]; a[[i]] = a[[p]]*a[[i/p]]], {i, 2, nn}]; a (* T. D. Noe, Dec 12 2004, revised Sep 27 2011 *)
Array[Apply[Times, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[ #]] /. p_ /; PrimeQ@ p :> PrimePi@ p + 1] &, 74] (* Michael De Vlieger, Aug 22 2017 *)

PARI

A064553(n)={n=factor(n);n[,1]=apply(f->1+primepi(f),n[,1]);factorback(n)} \\ M. F. Hasler, Aug 28 2012

Scheme

(define (A064553 n) (if (= 1 n) n (* (+ 1 (A055396 n)) (A064553 (A032742 n))))) ;; Antti Karttunen, Aug 22 2017

Formula

a(A000040(n)) = n+1.

Let the prime factorization of n be p1^e1...pk^ek, then a(n) = (pi(p1)+1)^e1...(pi(pk)+1)^ek, where pi(p) is the index of prime p. - T. D. Noe, Dec 12 2004

From Antti Karttunen, Aug 22 2017: (Start)

a(n) = A003963(A003961(n)).

a(A181819(n)) = A000005(n).

a(A290641(n)) = n. (End)

Extensions

Displayed values double-checked with new PARI code by M. F. Hasler, Aug 28 2012