A349658 -id:A349658 - cp's OEIS frontend

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

Wesley Ivan Hurt, Jul 07 2020

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

			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.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Refactorable Number

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

Cf. also A335182, A335665.

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

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

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

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

A335182 Sum of the refactorable divisors of n.

Original entry on oeis.org

1, 3, 1, 3, 1, 3, 1, 11, 10, 3, 1, 15, 1, 3, 1, 11, 1, 30, 1, 3, 1, 3, 1, 47, 1, 3, 10, 3, 1, 3, 1, 11, 1, 3, 1, 78, 1, 3, 1, 51, 1, 3, 1, 3, 10, 3, 1, 47, 1, 3, 1, 3, 1, 30, 1, 67, 1, 3, 1, 75, 1, 3, 10, 11, 1, 3, 1, 3, 1, 3, 1, 182, 1, 3, 1, 3, 1, 3, 1, 131, 10, 3, 1, 99

Offset: 1

Wesley Ivan Hurt, Jul 17 2020

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of refactorable numbers (A336040). - Wesley Ivan Hurt, Jun 21 2024

			a(6) = 3; The divisors of 6 are {1,2,3,6}. 1 and 2 are refactorable since d(1) = 1|1 and d(2) = 2|2, so a(6) = 1 + 2 = 3.
a(7) = 1; The divisors of 7 are {1,7} and 1 is the only refactorable divisor of 7. So a(7) = 1.
a(8) = 11; 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, so a(8) = 1 + 2 + 8 = 11.
a(9) = 10; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9, so a(9) = 1 + 9 = 10.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Refactorable Number

Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A336041 (number of refactorable divisors), A335665 (their product).

Difference of A349322 and A349658.

a[n_] := DivisorSum[n, # &, Divisible[#, DivisorSigma[0, #]] &]; Array[a, 80] (* Amiram Eldar, Nov 24 2021 *)

isr(n) = n%numdiv(n)==0; \\ A033950
a(n) = sumdiv(n, d, if (isr(d), d)); \\ Michel Marcus, Jul 20 2020

a(n) = Sum_{d|n} d * c(d), where c = A336040.
a(n) = Sum_{d|n} d * (1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).
a(n) = A349322(n) - A349658(n). - Antti Karttunen, Nov 24 2021
a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A349322 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 2, 12, 11, 5, 2, 18, 2, 5, 4, 13, 2, 32, 2, 7, 4, 5, 2, 50, 3, 5, 12, 7, 2, 9, 2, 14, 4, 5, 4, 81, 2, 5, 4, 55, 2, 9, 2, 7, 14, 5, 2, 52, 3, 7, 4, 7, 2, 34, 4, 71, 4, 5, 2, 83, 2, 5, 14, 15, 4, 9, 2, 7, 4, 9, 2, 185, 2, 5, 6, 7, 4, 9, 2, 136, 13, 5, 2, 107

Offset: 1

Wesley Ivan Hurt, Nov 14 2021

nonn

For each divisor d of n, add d if d is refactorable (i.e., if the number of divisors of d divides d), otherwise add 1. For example, the divisors of 8 are 1,2,4,8 and the refactorable divisors of 8 are 1,2,8. The sum is then a(8) = 1 + 2 + 1 + 8 = 12.

Inverse Möbius transform of n^c(n), where c = A336040. - Wesley Ivan Hurt, Jun 29 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Refactorable Number

Cf. A033950, A208251, A335182, A335665, A336040, A336041, A349658.

a[n_] := DivisorSum[n, If[Divisible[#, DivisorSigma[0, #]], #, 1] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

isrf(n) = n%numdiv(n)==0; \\ A336040
a(n) = sumdiv(n, d, if (isrf(d), d, 1)); \\ Michel Marcus, Nov 16 2021

a(n) = A335182(n) + A349658(n). - Antti Karttunen, Nov 24 2021

a(p) = 2 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

cp's OEIS Frontend

A336041 Number of refactorable divisors of n.

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A335182 Sum of the refactorable divisors of n.

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A349322 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).

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