A336041 Number of refactorable divisors of n.

1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 5, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 2, 1, 2, 1, 5, 2, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2

Offset

1,2

Comments

Inverse Möbius transform of A336040. - Antti Karttunen, Nov 24 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Eric Weisstein's World of Mathematics, Refactorable Number

Formula

a(n) = Sum_{d|n} c(d), where c(n) is the refactorable characteristic of n (A336040).

a(n) = Sum_{d|n} (1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).

a(n) = A000005(n) - A349658(n). - Antti Karttunen, Nov 24 2021

a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

Example

a(6) = 2; The divisors of 6 are {1,2,3,6}. Only two of these divisors are refactorable since d(1) = 1|1 and d(2) = 2|2, but d(3) = 2 does not divide 3 and d(6) = 4 does not divide 6.
a(7) = 1; The divisors of 7 are {1,7} and d(1) = 1|1, but d(7) = 2 does not divide 7, so a(7) = 1.
a(8) = 3; The divisors of 8 are {1,2,4,8}. 1, 2 and 8 are refactorable since d(1) = 1|1, d(2) = 2|2 and d(8) = 4|8 but d(4) = 3 does not divide 4, so a(8) = 3.
a(9) = 2; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9 but d(3) = 2 does not divide 3. Thus, a(9) = 2.

Maple

A336041 := proc(n)
    local a ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if type(d/numtheory[tau](d), integer) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A336041(n), n=1..30) ; # R. J. Mathar, Nov 24 2020

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, d%numdiv(d) == 0); \\ Michel Marcus, Jul 07 2020

Crossrefs

Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A349658 (number of nonrefactorable divisors).

Cf. also A335182, A335665.

Sequence in context: A167970 A126433 A237271 * A176725 A085029 A185318

Adjacent sequences: A336038 A336039 A336040 * A336042 A336043 A336044

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jul 07 2020

Status

approved