A116550 The bi-unitary analog of Euler's totient function of n.

1, 1, 2, 3, 4, 3, 6, 7, 8, 6, 10, 8, 12, 9, 9, 15, 16, 12, 18, 14, 14, 15, 22, 17, 24, 18, 26, 21, 28, 15, 30, 31, 23, 24, 25, 29, 36, 27, 28, 31, 40, 21, 42, 35, 34, 33, 46, 36, 48, 36, 37, 42, 52, 39, 42, 46, 42, 42, 58, 34, 60, 45, 51, 63, 50, 35, 66, 56, 51, 38, 70, 62, 72, 54

Offset

1,3

Comments

a(1)=1; for n>1, a(n) is the number of numbers m<n such that A225174(m,n)=1. - N. J. A. Sloane, May 01 2013

References

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Links

Donovan Johnson, Table of n, a(n) for n = 1..10000

László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS, Vol. 12 (2009), Article 09.5.2, function phi**(n).

Formula

For n>1, if n = product{p=primes,p|n} p^b(n,p), where each b(n,p) is a positive integer, then a(n) is number of positive integers m, m < n, such that each b(m,p) does not equal b(n,p).

a(n) = Sum_{d|n, gcd(d,n/d)=1} (-1)^omega(d) * phi(x, n), where phi(x, n) = #{1 <= k <= x, gcd(k, n) = 1} = Sum_{d|n} mu(d) * floor(x/d) (Tóth, 2009). - Amiram Eldar, Jul 16 2022

Example

12 = 2^2 * 3^1. Of the positive integers < 12, there are 8 integers where no prime divides these integers the same number of times the prime divides 12: 1, 2 = 2^1, 5 = 5^1, 7 = 7^1, 8 = 2^3, 9 = 3^2, 10 = 2^1 *5^1 and 11 = 11^1. So a(12) = 8. The other positive integers < 12 (3 = 3^1, 4 = 2^2 and 6 = 2^1 * 3^1) each are divisible by at least one prime the same number of times this prime divides 12.

Maple

# returns the greatest common unitary divisor of m and n, A225174(m, n)
f:=proc(m, n)
   local i, ans;
   ans:=1;
   for i from 1 to min(m, n) do
     if ((m mod i) = 0) and (igcd(i, m/i) = 1)  then
       if ((n mod i) = 0) and (igcd(i, n/i) = 1)  then ans:=i; fi;
     fi;
   od;
ans; end;
A116550:=proc(n)
  global f; local ct, m;
  ct:=0;
  if n = 1 then RETURN(1) else
  for m from 1 to n-1 do
    if f(m, n)=1 then ct:=ct+1; fi;
  od:
  fi;
  ct;
end; # N. J. A. Sloane, May 01 2013
A116550 := proc(n)
    local a, k;
    a := 0 ;
    for k from 1 to n do
        if A165430(k, n) = 1 then
            a := a+1 ;
        end if ;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 21 2016

Mathematica

a[1] = 1; a[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Sep 05 2013 *)
phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; a[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

Prog

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
a(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1)); \\ Michel Marcus, Nov 09 2017
(PARI) phi(x, n) = sumdiv(n, d, moebius(d)*floor(x/d));
a(n) = sumdiv(n, d, (gcd(d, n/d) == 1) * (-1)^omega(d) * phi(n/d, d)); \\ Amiram Eldar, Jul 16 2022

Crossrefs

Cf. A225174, A005424, A225175, A225176, A000010, A047994.

Sequence in context: A048276 A127463 A076618 * A283165 A116991 A330061

Adjacent sequences: A116547 A116548 A116549 * A116551 A116552 A116553

Keyword

nonn

Author

Leroy Quet, Mar 16 2006

Extensions

More terms from R. J. Mathar, Jan 23 2008

Entry revised by N. J. A. Sloane, May 01 2013

Status

approved