login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A336041 Number of refactorable divisors of n. 7
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 21 06:36 EDT 2022. Contains 353889 sequences. (Running on oeis4.)