A225174 -id:A225174 - cp's OEIS frontend

A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1
  1, 2
  1, 1, 3
  1, 1, 1, 4
  1, 1, 1, 1, 5
  1, 2, 3, 2, 1, 6
  1, 1, 1, 1, 1, 1, 7
  1, 1, 1, 1, 1, 1, 1, 8
  1, 1, 1, 1, 1, 1, 1, 1, 9
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10

Amiram Eldar, Table of n, a(n) for n = 1..10585 (first 145 rows flattened)

Cf. A005117, A050873, A077610, A145388 (row sums), A165430, A225174, A384046.

Upper right triangle of A322482.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; T[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
T(n, k) = vecmax(setintersect(udiv(n), divisors(k)));

T(n, 1) = 1.
T(n, n) = n.
T(n, k) <= A050873(n, k) = gcd(n, k), with equality if n is squarefree (A005117).

A384046 Triangle in which the n-th row gives the numbers from 1 to n whose largest divisor that is a unitary divisor of n is 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 3, 7, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 5, 7, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 3, 5, 9, 11, 13, 1, 2, 4, 7, 8, 11, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1,
  1,
  1, 2,
  1, 2, 3,
  1, 2, 3, 4,
  1, 5,
  1, 2, 3, 4, 5, 6,
  1, 2, 3, 4, 5, 6, 7,
  1, 2, 3, 4, 5, 6, 7, 8,
  1, 3, 7, 9

Amiram Eldar, Table of n, a(n) for n = 1..10204 (first 170 rows flattened)
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.

The unitary analog of A038566.

Cf. A047994 (row lengths), A333576 (row sums), A077610, A225174, A384047.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; uGCD[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; row[n_] := Select[Range[n], uGCD[n, #] == 1 &]; Array[row, 10] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
ugcd(n, k) = vecmax(setintersect(udiv(n), divisors(k)));
row(n) = select(x -> ugcd(n, x) == 1, vector(n, i, i));

T(n, 1) = 1.

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

Original entry on oeis.org

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

Leroy Quet, Mar 16 2006

nonn

a(1)=1; for n>1, a(n) is the number of numbers mA225174(m,n)=1. - N. J. A. Sloane, May 01 2013

			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.

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

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).

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

# 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

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 *)

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

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

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

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

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

A225175 Largest number which requires n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 12, 18, 30, 42, 78, 106, 210, 366, 550, 603, 750, 1290, 2562, 4398, 4305, 7470, 9090, 14322, 24558, 35382, 55482, 78020, 141190, 207519, 301642, 429870, 552693, 684846, 1060710, 1391390, 2385246, 3454044

Offset: 0

N. J. A. Sloane, May 01 2013

a(26) >= 55482. a(27) >= 78020. - R. J. Mathar, May 05 2013

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

Cf. A116550, A005424, A225176, A225174, A047994, A003271, A225172, A225173.

a(26)-a(37) from Donovan Johnson, Dec 07 2013

A225176 Number of numbers which require n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 2, 4, 7, 6, 13, 12, 16, 24, 31, 51, 66, 87, 126, 139, 187, 260, 331, 412, 551, 693

Offset: 1

N. J. A. Sloane, May 01 2013

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

Cf. A116550, A005424, A225175, A225174, A047994, A003271, A225172, A225173.

A384244 Triangle in which the n-th row gives the numbers k from 1 to n such that the greatest common unitary divisor of k and n is 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 3, 4, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 5, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 3, 4, 5, 8, 9, 11, 12, 13, 1, 2, 4, 7, 8, 9, 11, 13, 14

Offset: 1

Amiram Eldar, May 23 2025

			Triangle begins:
  1
  1
  1, 2
  1, 2, 3
  1, 2, 3, 4
  1, 4, 5
  1, 2, 3, 4, 5, 6
  1, 2, 3, 4, 5, 6, 7
  1, 2, 3, 4, 5, 6, 7, 8
  1, 3, 4, 7, 8, 9

Amiram Eldar, Table of n, a(n) for n = 1..10349 (first 160 rows flattened)
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.

The bi-unitary analog of A038566.

Cf. A116550 (row lengths), A200723 (row sums), A077610, A089912, A165430, A225174, A064379 (infinitary analog), A384046 (unitary analog).

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &];
ugcd[n_, m_] := Max[Intersection[udiv[n], udiv[m]]];
row[n_] := Select[Range[n], ugcd[n, #] == 1 &]; Array[row, 15] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
ugcd(n, m) = vecmax(setintersect(udiv(n), udiv(m)));
row(n) = select(x -> ugcd(n, x) == 1, vector(n, i, i));

cp's OEIS Frontend

A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

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A384046 Triangle in which the n-th row gives the numbers from 1 to n whose largest divisor that is a unitary divisor of n is 1.

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

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A225175 Largest number which requires n iterations of the bi-unitary totient function (A116550) to reach 1.

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Extensions

A225176 Number of numbers which require n iterations of the bi-unitary totient function (A116550) to reach 1.

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Author

Keywords

References

Crossrefs

A384244 Triangle in which the n-th row gives the numbers k from 1 to n such that the greatest common unitary divisor of k and n is 1.

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Programs

Mathematica

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