A033950 -id:A033950 - 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

A009230 a(n) = lcm(n, d(n)).

Original entry on oeis.org

1, 2, 6, 12, 10, 12, 14, 8, 9, 20, 22, 12, 26, 28, 60, 80, 34, 18, 38, 60, 84, 44, 46, 24, 75, 52, 108, 84, 58, 120, 62, 96, 132, 68, 140, 36, 74, 76, 156, 40, 82, 168, 86, 132, 90, 92, 94, 240, 147, 150, 204, 156, 106, 216, 220, 56, 228, 116, 118, 60, 122, 124, 126, 448

Offset: 1

David W. Wilson

nonn

Fixed points of this sequence are the refactorable numbers (A033950), i.e. a(A033950(n)) = A033950(n). - Labos Elemer, Nov 18 2002

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

Cf. A000005, A033950.

a:= n-> ilcm(n, numtheory[tau](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 14 2017

Table[LCM[n,DivisorSigma[0,n]],{n,70}] (* Harvey P. Dale, Jul 31 2016 *)

A281188 Number of refactorable numbers m such that tau(m) = n, or 0 if there are infinitely many such numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 6, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 6, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 6, 1, 0, 2, 6, 1, 0, 1, 2, 0, 0, 2, 6, 1, 0, 0, 2, 1, 0, 2, 2, 2, 0, 1, 0, 2, 0, 2, 2, 2, 0, 1, 0, 0, 0

Offset: 1

Franklin T. Adams-Watters, Jan 16 2017

An integer n is a refactorable number if and only if tau(n) (A000005) divides n.
Every number is tau(m) for some refactorable m.
If n is squarefree with k prime divisors, then a(n) = k! (for a proof, see the Links entry from the author).
Conjecture: a(n) is nonzero if and only if n is squarefree or n = 4. [This conjecture is true; see Links for a proof. - Jon E. Schoenfield and Altug Alkan, Jan 17 2017]
See also Theorem 5 for the proof of conjecture in Colton link. - Altug Alkan, Jan 20 2017

			If n is prime, the only refactorable number m with tau(m) = n is n^(n-1), so a(n) = 1 for n prime.
Any number n of the form 8p, p a prime not equal to 2, has tau(n) = 8, and thus n is refactorable. Hence a(8) = 0.

Altug Alkan, Table of n, a(n) for n = 1..10000
Franklin T. Adams-Watters, Refactorable numbers with tau squarefree
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
Jon E. Schoenfield and Altug Alkan, Refactorable numbers with tau nonsquarefree

Cf. A000005, A033950, A039819.

k = 1; t[] = 0; t[4] = 1; While[k < 100000001, m = DivisorSigma[0, k]; If[ Mod[k, m] == 0 && SquareFreeQ@ m, t[m]++]; k++]; t@# & /@ Range@20 (* _Robert G. Wilson v, Jan 16 2017 *)

a(n) = if(n==4, 1, if(issquarefree(n) == 1, omega(n)!, 0)); \\ Altug Alkan, Jan 18 2017

More terms from Altug Alkan, Jan 17 2017

A057532 n is odd and sum of digits of n equals the numbers of divisors of n.

Original entry on oeis.org

1, 11, 101, 225, 301, 441, 525, 1003, 1111, 1425, 1521, 1575, 1911, 2015, 2101, 2325, 2475, 2541, 2601, 2925, 3225, 3311, 3825, 4275, 4301, 4851, 5025, 5175, 5445, 5733, 5775, 6075, 6321, 6525, 7315, 7605, 7623, 8325, 8925, 9225, 9555, 10003, 10021

Offset: 1

Asher Auel, Sep 03 2000

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A007953, A033950, A057531.

Select[Range[1, 10^4, 2], Plus @@ IntegerDigits[#] == DivisorSigma[0, #] &] (* Giovanni Resta, Apr 24 2017 *)

A245782 Refactorable multiply-perfect numbers.

Original entry on oeis.org

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Jaroslav Krizek, Aug 01 2014

nonn

Multiply-perfect numbers k (A007691) such that k / tau(k) is an integer.
Also multiply-perfect numbers k (A007691) such that (k / tau(k) - sigma(k) / k) = (k / A000005(k) - A000203(k) / k) is an integer.
Also multiply-perfect numbers k (A007691) such that (k / tau(k) + sigma(k) / k) = (k / A000005(k) + A000203(k) / k) is an integer.

			Multiply-perfect number 672 is in sequence because 672 / tau(672) = 28 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..407

Intersection of A033950 (refactorable numbers) and A007691 (multiply-perfect numbers).
Subsequence of A245778 and A245786.
Supersequence of A047728.
Cf. A000005, A000203.
Cf. A245776, A245777, A245779.

[n:n in [A007691(n)] | (Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) eq 1];

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

isok(n) = !(n % numdiv(n)) && !(sigma(n) % n); \\ Michel Marcus, Aug 11 2014

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !(k % d);} \\ Amiram Eldar, May 09 2024

a(14)-a(18) from Amiram Eldar, May 09 2024

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

A336064 Numbers divisible by the maximal exponent in their prime factorization (A051903).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is A336065 = 0.848957... (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051903(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051903, A124184, A336065.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336063.

H[1] = 0; H[n_] := Max[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, H[#]] &]

isok(m) = if (m>1, (m % vecmax(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

A046754 Numbers k such that the square of d(k) (number of divisors) divides k.

Original entry on oeis.org

1, 9, 128, 625, 972, 1152, 2000, 2025, 5625, 6561, 7776, 8100, 10000, 10800, 18000, 21952, 26244, 30000, 32768, 35721, 50625, 55296, 56700, 64000, 64800, 65856, 70000, 80000, 84672, 89100, 90000, 97200, 98304, 99225, 105300, 109760, 110000

Offset: 1

Labos Elemer

nonn

Subset of A033950.

			If k = 972, d(k) = sigma(0,k) = 18. Its square is 324 which divides 972.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A033950, A046755, A046756.

Select[Range[110000], IntegerQ[#/DivisorSigma[0, #]^2] &] (* Jayanta Basu, Jun 28 2013 *)

A272872 Numbers k such that k+1 is divisible by number of divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 123, 127, 131, 135, 137, 139, 143, 149, 151, 155, 157, 159, 163, 167

Offset: 1

Altug Alkan, May 11 2016

nonn

Inspired by A272353.
All odd primes are obvious members.
Numbers k such that k == -1 (mod A000005(k)). Nonprime terms are listed in A354714. - Max Alekseyev, Jun 04 2022
63 is the least number that is not in this sequence but is a member of A187929.

			15 is a term because A000005(15) = 4 divides 15+1 = 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia A. Spiro, How Often Does the Number of Divisors of an Integer Divide Its Successor?, J. London Math. Soc. (1985) s2-31 (1): 30-40.

Cf. A000005, A187929, A272353, A033950, A354711, A354712, A354714, A354715, A354716.

Select[Range@167, Mod[#+1, DivisorSigma[0, #]] == 0 &] (* Giovanni Resta, May 21 2016 *)

lista(nn) = {for(n=1, nn, if((n+1) % numdiv(n) == 0, print1(n, ", ")));}

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

cp's OEIS Frontend

A336041 Number of refactorable divisors of n.

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Maple

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A009230 a(n) = lcm(n, d(n)).

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A281188 Number of refactorable numbers m such that tau(m) = n, or 0 if there are infinitely many such numbers.

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A057532 n is odd and sum of digits of n equals the numbers of divisors of n.

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A245782 Refactorable multiply-perfect numbers.

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Magma

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A335182 Sum of the refactorable divisors of n.

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A336064 Numbers divisible by the maximal exponent in their prime factorization (A051903).

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