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

%I A064553 #58 Nov 21 2021 07:39:29
%S A064553 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,
%T A064553 20,11,24,12,32,18,16,20,36,13,18,21,32,14,30,15,24,36,20,16,48,25,32,
%U A064553 24,28,17,54,24,40,27,22,18,48,19,24,45,64,28,36,20,32,30,40,21,72,22,26
%N A064553 a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.
%C A064553 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.
%C A064553 Completely multiplicative with a(prime(i)) = i + 1. - _Charles R Greathouse IV_, Sep 07 2012
%C A064553 a(A080688(n,k)) = A080444(n,k) = n for k=1..A001055(n). - _Reinhard Zumkeller_, Oct 01 2012
%H A064553 T. D. Noe, <a href="/A064553/b064553.txt">Table of n, a(n) for n = 1..8000</a>
%H A064553 T. D. Noe, <a href="/A064553/a064553.gif">Plot of A064553</a>
%H A064553 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A064553 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A064553 a(A000040(n)) = n+1.
%F A064553 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
%F A064553 From _Antti Karttunen_, Aug 22 2017: (Start)
%F A064553 a(n) = A003963(A003961(n)).
%F A064553 a(A181819(n)) = A000005(n).
%F A064553 a(A290641(n)) = n. (End)
%e A064553 a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.
%p A064553 A064553 := proc(n)
%p A064553     local a,f,p,e ;
%p A064553     a := 1 ;
%p A064553     for f in ifactors(n)[2] do
%p A064553         p :=op(1,f) ;
%p A064553         e :=op(2,f) ;
%p A064553         a := a*(numtheory[pi](p)+1)^e ;
%p A064553     end do:
%p A064553     a ;
%p A064553 end proc: # _R. J. Mathar_, Sep 07 2012
%t A064553 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 *)
%t A064553 Array[Apply[Times, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[ #]] /. p_ /; PrimeQ@ p :> PrimePi@ p + 1] &, 74] (* _Michael De Vlieger_, Aug 22 2017 *)
%o A064553 (Haskell)
%o A064553 a064553 1 = 1
%o A064553 a064553 n = product $ map ((+ 1) . a049084) $ a027746_row n
%o A064553 -- _Reinhard Zumkeller_, Apr 09 2012, Feb 17 2012, Jan 28 2011
%o A064553 (PARI) A064553(n)={n=factor(n);n[,1]=apply(f->1+primepi(f),n[,1]);factorback(n)} \\ _M. F. Hasler_, Aug 28 2012
%o A064553 (Scheme) (define (A064553 n) (if (= 1 n) n (* (+ 1 (A055396 n)) (A064553 (A032742 n))))) ;; _Antti Karttunen_, Aug 22 2017
%Y A064553 Cf. A000005, A000040, A003961, A003963, A049084, A020639, A064554, A064555, A001055, A003586, A064557, A064558, A027746, A027748, A124010, A181819.
%Y A064553 A left inverse of A290641.
%K A064553 mult,nice,nonn,look
%O A064553 1,2
%A A064553 _Reinhard Zumkeller_, Sep 21 2001
%E A064553 Displayed values double-checked with new PARI code by _M. F. Hasler_, Aug 28 2012