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
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