OFFSET
1,2
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
LINKS
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(A290641(n)) = n. (End)
EXAMPLE
a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.
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 *)
PROG
(Haskell)
a064553 1 = 1
a064553 n = product $ map ((+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Feb 17 2012, Jan 28 2011
(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
CROSSREFS
KEYWORD
AUTHOR
Reinhard Zumkeller, Sep 21 2001
EXTENSIONS
Displayed values double-checked with new PARI code by M. F. Hasler, Aug 28 2012
STATUS
approved