A336040 -id:A336040 - cp's OEIS frontend

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

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

A336041 Number of refactorable divisors of n.

Original entry on oeis.org

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

A335665 Product of the refactorable divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 16, 9, 2, 1, 24, 1, 2, 1, 16, 1, 324, 1, 2, 1, 2, 1, 4608, 1, 2, 9, 2, 1, 2, 1, 16, 1, 2, 1, 139968, 1, 2, 1, 640, 1, 2, 1, 2, 9, 2, 1, 4608, 1, 2, 1, 2, 1, 324, 1, 896, 1, 2, 1, 1440, 1, 2, 9, 16, 1, 2, 1, 2, 1, 2, 1, 1934917632, 1, 2, 1, 2, 1, 2, 1, 51200

Offset: 1

Wesley Ivan Hurt, Jul 17 2020

			a(6) = 2; 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 = 2.
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) = 16; 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 = 16.
a(9) = 9; 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 = 9.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number

Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A336041 (number of refactorable divisors), A335182 (their sum).

Cf. also A349322 (similar formula, but with sum instead of product).

a[n_] := Product[If[Divisible[d, DivisorSigma[0, d]], d, 1], {d, Divisors[n]}]; Array[a, 60] (* Amiram Eldar, Nov 24 2021 *)

isr(n) = n%numdiv(n)==0; \\ A033950
a(n) = my(d=divisors(n)); prod(k=1, #d, if (isr(d[k]), d[k], 1)); \\ Michel Marcus, Jul 18 2020

a(n) = Product_{d|n} d^c(d), where c(n) is the refactorable characteristic of n (A336040).
a(n) = Product_{d|n} d^(1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).
a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A208251 Number of refactorable numbers less than or equal to n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Wesley Ivan Hurt, Jan 12 2013

A number is refactorable if it is divisible by the number of its divisors.

			a(1) = 1 since 1 is the first refactorable number, a(2) = 2 since there are two refactorable numbers less than or equal to 2, a(3) through a(7) = 2 since the next refactorable number is 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number.

Cf. A033950, A000005, A141586, A057265, A036896, A036898, A114617.

Partial sums of A336040.

with(numtheory) a:=n->sum((1 + floor(i/tau(i)) - ceil(i/tau(i))), i=1..n);

Accumulate[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 1,100}]] (* Amiram Eldar, Oct 11 2023 *)

a(n) = sum(i=1, n, q = i/numdiv(i); 1+ floor(q) - ceil(q)); \\ Michel Marcus, Sep 10 2018

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

A368625 Characteristic function of non-refactorable numbers (A159973).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1

Offset: 1

Wesley Ivan Hurt, Jan 01 2024

Antti Karttunen, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Refactorable Number
Index entries for characteristic functions

Cf. A000005, A033950 (refactorable numbers), A054008, A159973, A336040.

Table[(Ceiling[n/DivisorSigma[0, n]] - Floor[n/DivisorSigma[0, n]]), {n, 100}]

A368625(n) = !!(n%numdiv(n)); \\ Antti Karttunen, Jan 17 2025

a(n) = ceiling(n/d(n))-floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
a(n) = 1 - A336040(n).
a(n) = [A054008(n) > 0], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 17 2025

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A368572 Number of refactorable numbers less than n that do not divide n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 3, 2, 4, 3, 4, 2, 4, 2, 5, 4, 5, 4, 5, 2, 6, 5, 5, 5, 6, 5, 6, 4, 6, 5, 6, 2, 7, 6, 7, 5, 8, 7, 8, 7, 7, 7, 8, 4, 8, 7, 8, 7, 8, 5, 8, 6, 9, 8, 9, 7, 10, 9, 9, 8, 10, 9, 10, 9, 10, 9, 10, 3, 11, 10, 11, 10, 11, 10, 11, 8, 11, 11, 12, 10

Offset: 1

Wesley Ivan Hurt, Jan 06 2024

			a(15) = 4 since there are 4 refactorable numbers that are less than 15 and do not divide 15, namely: 2, 8, 9, 12.

Cf. A033950, A336040, A368817.

Table[Sum[(1 - Ceiling[k/DivisorSigma[0, k]] + Floor[k/DivisorSigma[0, k]]) (Ceiling[n/k] - Floor[n/k]), {k, n}], {n, 100}]

a(n) = Sum_{k=1..n} c(k) * (ceiling(n/k) - floor(n/k)), where c = A336040.

A368817 Sum of the refactorable numbers less than n that do not divide n.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 0, 10, 17, 19, 17, 31, 29, 31, 21, 31, 20, 49, 47, 49, 47, 49, 27, 73, 71, 64, 71, 73, 71, 73, 63, 73, 71, 73, 32, 109, 107, 109, 99, 149, 147, 149, 147, 140, 147, 149, 103, 149, 147, 149, 147, 149, 120, 149, 139, 205, 203, 205, 191, 265, 263

Offset: 1

Wesley Ivan Hurt, Jan 06 2024

			a(15) = 31. There are 4 refactorable numbers that are less than 15 that do not divide 15, namely: 2, 8, 9, 12. Their sum is 2 + 8 + 9 + 12 = 31.

Cf. A033950, A336040, A368572.

Table[Sum[k (1 - Ceiling[k/DivisorSigma[0, k]] + Floor[k/DivisorSigma[0, k]]) (Ceiling[n/k] - Floor[n/k]), {k, n}], {n, 100}]

a(n) = Sum_{k=1..n} k * c(k) * (ceiling(n/k) - floor(n/k)), where c = A336040.

A368821 Number of compositions of n into 2 refactorable parts.

Original entry on oeis.org

0, 1, 2, 1, 0, 0, 0, 0, 2, 4, 2, 0, 2, 2, 0, 1, 2, 1, 2, 4, 2, 0, 0, 1, 2, 4, 2, 0, 0, 2, 0, 2, 2, 0, 0, 3, 2, 2, 0, 0, 2, 4, 0, 2, 2, 0, 0, 5, 2, 0, 0, 2, 0, 2, 0, 0, 2, 4, 0, 2, 2, 2, 0, 4, 2, 0, 0, 4, 2, 0, 0, 3, 2, 4, 0, 2, 0, 2, 0, 5, 4, 2, 0, 4, 2, 2, 0, 2, 4, 4, 0, 6, 2, 0, 0, 10, 4

Offset: 1

Wesley Ivan Hurt, Jan 06 2024

			a(10) = 4. There are 4 ordered ways to write 10 as the sum of two refactorable numbers: 1 + 9 = 2 + 8 = 8 + 2 = 9 + 1.
a(36) = 3. There are 3 ordered ways to write 36 as the sum of two refactorable numbers: 12 + 24 = 18 + 18 = 24 + 12.

Index entries for sequences related to compositions

Cf. A033950, A336040.

Table[Sum[(1 - Ceiling[k/DivisorSigma[0, k]] + Floor[k/DivisorSigma[0, k]]) (1 - Ceiling[(n - k)/DivisorSigma[0, (n - k)]] + Floor[(n - k)/DivisorSigma[0, (n - k)]]), {k, n - 1}], {n, 100}]

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A336040.

cp's OEIS Frontend

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

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A336041 Number of refactorable divisors of n.

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A335182 Sum of the refactorable divisors of n.

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A335665 Product of the refactorable divisors of n.

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A208251 Number of refactorable numbers less than or equal to n.

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A368625 Characteristic function of non-refactorable numbers (A159973).

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A368572 Number of refactorable numbers less than n that do not divide n.

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