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

1,3

Comments

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

Links

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

Formula

a(n) = Sum_{d|tau(n)} sigma(tau(n)/d)*mu(d)*gcd(d,n). - Ridouane Oudra, Jun 17 2026

Example

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.

Maple

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

Mathematica

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

Prog

(PARI) 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
(Python)
from math import prod
from collections import Counter
from sympy import factorint
def A137927(n):
    f = factorint(n)
    q = set(f)
    return prod(p**e for p, e in sum((Counter(factorint(d+1)) for d in f.values()), start=Counter()).items() if p not in q) # Chai Wah Wu, Jun 17 2026

Crossrefs

Cf. A137926, A007955, A120736, A291899, A000005, A000203, A008683.

Sequence in context: A316657 A277914 A126626 * A321864 A084311 A353486

Adjacent sequences: A137924 A137925 A137926 * A137928 A137929 A137930

Keyword

nonn,changed

Author

Leroy Quet, Feb 23 2008

Extensions

More terms from Stefan Steinerberger, Mar 09 2008

Status

approved