A137926 -id:A137926 - cp's OEIS frontend

A137927 a(n) = the largest divisor of A000005(n) that is coprime to n. (A000005(n) = the number of positive divisors of n.).

1, 1, 2, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 5, 2, 1, 2, 3, 4, 1, 2, 1, 3, 1, 4, 3, 2, 1, 2, 3, 4, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 3, 2, 1, 2, 5, 3, 3, 4, 3, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 2, 7, 4, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1

Offset: 1

Leroy Quet, Feb 23 2008

nonn

Apparently also the denominator of A007955(n)/A000005(n). See A291186. - Jaroslav Krizek, Sep 05 2017

			20 has 6 positive divisors. The divisors of 6 are 1,2,3,6. The divisors of 6 that are coprime to 20 are 1 and 3. 3 is the largest of these; so a(20) = 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A137926, A007955, A120736, A291899.

A137927 := proc(n)
    local a;
    a := 1 ;
    for d in numtheory[divisors](numtheory[tau](n)) do
        if igcd(d,n) = 1 then
            a := max(a,d) ;
        end if:
    end do:
    a ;
end proc:
seq(A137927(n),n=1..100) ; # R. J. Mathar, Sep 22 2017

Table[Select[Divisors[Length[Divisors[n]]], GCD[ #, n] == 1 &][[ -1]], {n, 1, 80}] (* Stefan Steinerberger, Mar 09 2008 *)

a(n) = my(d=divisors(numdiv(n))); forstep(k=#d, 1, -1, if (gcd(d[k], n) == 1, return (d[k]))); \\ Michel Marcus, Sep 22 2017; corrected Jun 13 2022

More terms from Stefan Steinerberger, Mar 09 2008

A152442 n is included if the largest divisor of n that is coprime to d(n) is a composite, where d(n) is the number of divisors of n.

Original entry on oeis.org

4, 15, 16, 21, 25, 27, 30, 33, 35, 36, 39, 42, 49, 50, 51, 54, 55, 57, 64, 65, 66, 69, 70, 75, 77, 78, 81, 85, 87, 91, 93, 95, 98, 100, 102, 105, 110, 111, 114, 115, 119, 120, 121, 123, 125, 129, 130, 133, 135, 138, 140, 141, 143, 144, 145, 147, 150, 154, 155, 159, 161

Offset: 1

Leroy Quet, Dec 04 2008

nonn

A137926(a(n)) is composite for all n.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A137926, A152444.

a = {}; For[n = 2, n < 1000, n++, b = Max[Select[Divisors[n], GCD[DivisorSigma[0, n], # ] == 1 &]]; If[b > 1 && ! PrimeQ[b], AppendTo[a, n]]]; a (* Stefan Steinerberger, Dec 06 2008 *)
ldcQ[n_]:=CompositeQ[Max[Select[Divisors[n],CoprimeQ[#,DivisorSigma[0,n]]&]]]; Select[Range[200],ldcQ] (* Harvey P. Dale, Feb 19 2025 *)

More terms from Stefan Steinerberger and Ray Chandler, Dec 06 2008

A152443 a(n) = the largest proper divisor of n that is coprime to d(n), where d(n) = the number of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 1, 7, 5, 8, 1, 1, 1, 5, 7, 11, 1, 3, 5, 13, 9, 7, 1, 15, 1, 1, 11, 17, 7, 4, 1, 19, 13, 5, 1, 21, 1, 11, 5, 23, 1, 3, 7, 25, 17, 13, 1, 27, 11, 7, 19, 29, 1, 5, 1, 31, 7, 32, 13, 33, 1, 17, 23, 35, 1, 1, 1, 37, 25, 19, 11, 39, 1, 1, 27, 41, 1, 7, 17, 43, 29, 11, 1, 5

Offset: 2

Leroy Quet, Dec 04 2008

If n is squarefree then a(n) = A032742(n). - Robert Israel, Jun 23 2025

			There are 4 divisors of 27. These are 1,3,9,27. The proper divisors of 27 are therefore 1,3,9. The largest of these that is coprime to d(27) = 4 is 9. So a(27) = 9. (Note that in this case, the largest divisor of 27 coprime to 4 is 27, not 9.)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000005, A032742, A137926, A152444.

f:= proc(n) local d,D;
   D:= numtheory:-divisors(n);
   d:= nops(D);
   max(select(t -> igcd(d,t)=1, D minus {n}))
end proc:
map(f, [$2..100]); # Robert Israel, Jun 23 2025

Table[Max[Select[Drop[Divisors[n], -1], GCD[DivisorSigma[0, n], # ] == 1 &]], {n, 2, 100}] (* Stefan Steinerberger, Dec 06 2008 *)

Extended by Ray Chandler and Stefan Steinerberger, Dec 05 2008

cp's OEIS Frontend

A137927 a(n) = the largest divisor of A000005(n) that is coprime to n. (A000005(n) = the number of positive divisors of n.).

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A152442 n is included if the largest divisor of n that is coprime to d(n) is a composite, where d(n) is the number of divisors of n.

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A152443 a(n) = the largest proper divisor of n that is coprime to d(n), where d(n) = the number of divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions