A061537 -id:A061537 - cp's OEIS frontend

A062509 a(n) = n^omega(n).

1, 2, 3, 4, 5, 36, 7, 8, 9, 100, 11, 144, 13, 196, 225, 16, 17, 324, 19, 400, 441, 484, 23, 576, 25, 676, 27, 784, 29, 27000, 31, 32, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 74088, 43, 1936, 2025, 2116, 47, 2304, 49, 2500, 2601, 2704, 53, 2916

Offset: 1

Labos Elemer, Jul 13 2001

nonn

Not always equal to product of unitary divisors of n [compare with A061537]. This deviates from A061537 at 30, 42, 60, 66, etc.

			n=30: a(30) = 30^3 = 27000;
n=72: a(72) = 72^2 = 5184.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A061537, A034444, A000005, A033992.

Cf. A176029, A229109.

a062509 n = n ^ a001221 n  -- Reinhard Zumkeller, Sep 13 2013

Table[n^PrimeNu[n],{n,60}] (* Harvey P. Dale, Jan 14 2013 *)

a(n)={n^omega(n)} \\ Harry J. Smith, Aug 08 2009

a(n) = Sum_{d|n} tau(d^n)*mu(n/d). - Ridouane Oudra, Sep 17 2022

A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 72, 13, 14, 15, 128, 17, 108, 19, 200, 21, 22, 23, 144, 125, 26, 81, 392, 29, 30, 31, 64, 33, 34, 35, 46656, 37, 38, 39, 400, 41, 42, 43, 968, 675, 46, 47, 3456, 343, 500, 51, 1352, 53, 324, 55, 784, 57, 58, 59, 1800, 61, 62, 1323, 4096

Offset: 1

Jaroslav Krizek, Mar 01 2009

nonn

The exponential divisors of a number n = Product p(i)^e(i) are all numbers of the form Product p(i)^s(i) where s(i) divides e(i) for all i.

Not multiplicative: a(3)=3 (e-divisor 3^1), a(4)=8 (e-divisors 2^1 and 2^2), but a(12)=72 (e-divisors 3*2 and 3*2^2) <> a(3)*a(4). - R. J. Mathar, Apr 14 2011

			For n = 16 = 2^4 = the a(16) = 2^(A000203(4)) = 2^7 = 128. e-divisors of number 16 is 2, 4, 16, their product is 128.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, A note on exponential divisors and related arithmetic functions, Scientia Magna, Vol. 1, No. 1 (2005), pp. 97-101.

Cf. A000005, A000027, A000040, A000203, A006881, A000961, A049419, A051377, A120944, A322791.

Similar sequences: A007955, A061537, A274029.

[ &*[ d: d in Divisors(n) | forall(t) { p: p in P | v gt 0 and e mod v eq 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..64] ]; // Klaus Brockhaus, May 26 2009

f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] :=(Times @@ (f @@@ (fct = FactorInteger[n])))^(Times @@ DivisorSigma[0, Last /@ fct]); Array[a, 100] (* Amiram Eldar, Jun 03 2020 *)

a(1) = 1, a(p) = p, a(p*q) = p*q, a(p*q...*z) = pq...z, a(p^k) = p^(A000203(k)), for p, q, ..., z distinct primes and k > 1 an integer.
From Amiram Eldar, Jun 03 2020: (Start)
If n = Product_{i} p_i^e_i then a(n) = Product_{i} p_i^(sigma(e_i) * d_exp(n) / d(e_i)), where d_exp(n) = Product_{i} d(e_i) is the number of exponential divisors of n (A049419), d(e) and sigma(e) are the number of divisors (A000005) of e and their sum (A000203).
a(n) <= A007955(n) with equality if and only if n is noncomposite. (End)

a(1) = 1 from N. J. A. Sloane, Mar 02 2009

a(60) corrected by Klaus Brockhaus, May 26 2009

A290480 Product of proper unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 27000, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 74088, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 216000, 1, 62, 63, 1, 65, 287496, 1, 68, 69, 343000, 1, 72, 1, 74, 75, 76, 77, 474552, 1, 80

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(12) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are proper unitary {1, 3, 4} and 1*3*4 = 12.

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

Cf. A001221, A007774 (fixed points), A007955, A007956, A034460, A061537, A062758, A063919, A078599, A087652, A126192, A136655, A183091.

with(numtheory):
a:= n-> mul(d, d=select(x-> igcd(x, n/x)=1, divisors(n) minus {n})):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 03 2017

Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]

A290480(n) = if(1==n,n,n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d) == 1)//n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

a(n) = A061537(n)/n.
a(n) = n^(2^(omega(n)-1)-1), where omega() is the number of distinct primes dividing n (A001221).
a(n) = 1 if n is a prime power.

A062513 Product of unitary divisors of n is divided by n^(number of distinct prime factors).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 60, 1, 1, 1, 1, 1, 66, 1, 1, 1, 70, 1, 1, 1, 1, 1, 1, 1, 78, 1, 1, 1, 1, 1, 84, 1, 1, 1, 1, 1, 90, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Jul 13 2001

nonn

			n=210, with 4 p-divisors; all its 16 divisors are unitary; product=210^(16/2)=3782285936100000000, while 210^4=1944810000; a(210)=3782285936100000000/1944810000=1944810000.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A034444, A001221.

Table[n^(2^(PrimeNu[n] - 1) - PrimeNu[n]), {n,1,50}] (* G. C. Greubel, May 20 2017 *)

for(n=1,50, print1(round(n^(2^(omega(n) -1) - omega(n))), ", ")) \\ G. C. Greubel, May 20 2017

a(n) = A061537(n)/[n^A001221(n)].

a(n) = n^[(A034444(n)/2)-A001221(n)].

A064030 Product of unitary divisors of n!.

Original entry on oeis.org

1, 2, 36, 576, 207360000, 268738560000, 416336312719673760153600000000, 6984964247141514123629140377600000000, 300679807141675805997423113304381849600000000

Offset: 1

Labos Elemer, Sep 13 2001

nonn

			n = 6 has 8 unitary divisors:{16,45,9,80,5,144,1,720}, a(6) = 720^4 = 268738560000

Amiram Eldar, Table of n, a(n) for n = 1..17

Cf. A048656, A048657, A061537, A061538, A000142.

a[n_] := (n!)^(2^(PrimePi[n]-1)); Array[a, 10] (* Amiram Eldar, Jul 16 2019 *)

a(n)=(n!)^(A034444(n!)/2)

A064031 Product of non-unitary divisors of n!.

Original entry on oeis.org

1, 1, 1, 576, 207360000, 26956124946896309452800000000000, 2841003716170671644367609186370356458508919205193278721884160000000000000000000000

Offset: 1

Labos Elemer, Sep 13 2001

nonn

			n=6: 720 has 22 non-unitary divisors: a(6)=720^11=26956124946896309452800000000000

Amiram Eldar, Table of n, a(n) for n = 1..10

Cf. A048656, A048657, A061537, A061538, A000142, A048105, A000005, A034444.

a[n_] := (n!)^(DivisorSigma[0, n!]/2 - 2^(PrimePi[n]-1)); Array[a, 10] (* Amiram Eldar, Jul 16 2019 *)

a(n) = A061538(n!) = (n!)^(A048105(n!)/2) = (n!)^((A000005(n!)-A034444(n!))/2)

A290479 Product of nonprime squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 27000, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 74088, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 27000, 1, 62, 21, 1, 65, 287496, 1, 34, 69, 343000, 1, 6, 1, 74, 15, 38, 77, 474552, 1, 10

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(30) = 27000 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} and 1*6*10*15*30 = 27000.

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

Cf. A000005, A000469, A006881 (fixed points), A007947, A007955, A007956, A061537, A062758, A078599, A087652, A126192, A136655, A183091, A284118.

Table[Product[d, {d, Select[Divisors[n], !PrimeQ[#] && SquareFreeQ[#] &]}], {n, 80}]
Table[Last[Select[Divisors[n], SquareFreeQ]]^(DivisorSigma[0, Last[Select[Divisors[n], SquareFreeQ]]]/2 - 1), {n, 80}]

A290479(n) = if(1==n, n, my(r=factorback(factorint(n)[, 1])); (r^((numdiv(r)/2)-1))); \\ Antti Karttunen, Aug 06 2018

a(n) = A078599(n)/A007947(n).
a(n) = rad(n)^(d(rad(n))/2-1), where d() is the number of divisors of n (A000005) and rad() is the squarefree kernel of n (A007947).
a(n) = 1 if n is a prime power.

A384763 Product of the Euler totients of the unitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 4, 6, 16, 10, 16, 12, 36, 64, 8, 16, 36, 18, 64, 144, 100, 22, 64, 20, 144, 18, 144, 28, 4096, 30, 16, 400, 256, 576, 144, 36, 324, 576, 256, 40, 20736, 42, 400, 576, 484, 46, 256, 42, 400, 1024, 576, 52, 324, 1600, 576, 1296, 784, 58, 65536

Offset: 1

Darío Clavijo, Jun 09 2025

nonn

a(n) is the product of phi(d) over all unitary divisors d of n; i.e., those divisors satisfying gcd(d, n/d) = 1.
a(n) is upper bounded by A061537(n) (product of phi(d) over all divisors d of n).
The function is not multiplicative.
The sum of the totients over all unitary divisors d of n is A055653(n).

			For n = 6, a(6) = phi(1) * phi(2) * phi(3) * phi(6) = 1*1*2*2 = 4.

Cf. A000010, A034444, A055653, A061537, A077610.

a[n_] := EulerPhi[n]^(2^(PrimeNu[n] - 1)); Array[a, 100] (* Amiram Eldar, Jun 09 2025 *)

a(n) = my(p=1); fordiv(n, d, if (gcd(d,n/d) == 1, p*=eulerphi(d))); p; \\ Michel Marcus, Jun 09 2025

from sympy import totient, divisors, gcd
def a(n):
   prod = 1
   for d in divisors(n):
      if gcd(d, n//d) == 1:
          prod *= totient(d)
   return prod
print([a(n) for n in range(1, 61)])

a(n) = Product_{d|n} phi(d) if gcd(n,floor(n/d)) = 1.
a(p) = p-1 for p prime.
a(p^k) = p^k-p^(k-1).
a(n) = phi(n)^(2^(omega(n)-1)) = A000010(n)^(A034444(n)/2). - Amiram Eldar, Jun 09 2025

A064032 Product of unitary divisors of binomial(n, floor(n/2)).

Original entry on oeis.org

1, 2, 3, 36, 100, 400, 1225, 24010000, 252047376, 4032758016, 2075562447064149770496, 531343986448422341246976, 75186222935463997063888896, 19247673071478783248355557376, 2940278105018015412903875390625, 566574142904620264536665169363475932852029446342410000000000000000

Offset: 1

Labos Elemer, Sep 13 2001

nonn

Cf. A001405, A048243, A056173, A061537, A061538.

f[n_] := n^(2^(PrimeNu[n]-1)); Table[f[Binomial[n, Floor[n/2]]], {n, 1, 20}] (* Amiram Eldar, Jul 22 2024 *)

a(n) = apply(x -> x^(2^(omega(x)-1)), binomial(n, n\2)); \\ Amiram Eldar, Jul 22 2024

a(n) = A061537(A001405(n)). - Amiram Eldar, Jul 22 2024

a(15)-a(16) from Amiram Eldar, Jul 22 2024

A064033 Product of non-unitary divisors of binomial(n, floor(n/2)) or a(n) = 1 if all divisors are unitary. See A046098.

Original entry on oeis.org

1, 1, 1, 1, 1, 20, 1, 1, 15876, 1016255020032, 1, 728933458176, 8670998958336, 19247673071478783248355557376, 1714723915100625, 752711194884611945703392100000000, 1, 31226235883841773375939805209600000000, 1, 1357651828905889565182743230460164655087616

Offset: 1

Labos Elemer, Sep 13 2001

nonn

Cf. A001405, A048243, A056173, A061537, A061538.

f[n_] := n^((DivisorSigma[0, n] - 2^PrimeNu[n]) / 2); Table[f[Binomial[n, Floor[n/2]]], {n, 1, 20}] (* Amiram Eldar, Jul 22 2024 *)

a(n) = apply(x -> x^((numdiv(x) - 2^omega(x))/2), binomial(n, n\2)); \\ Amiram Eldar, Jul 22 2024

a(n) = A061538(A001405(n)).

a(18)-a(20) from Amiram Eldar, Jul 22 2024

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A062509 a(n) = n^omega(n).

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A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

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A290480 Product of proper unitary divisors of n.

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A062513 Product of unitary divisors of n is divided by n^(number of distinct prime factors).

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A064030 Product of unitary divisors of n!.

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A064031 Product of non-unitary divisors of n!.

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A290479 Product of nonprime squarefree divisors of n.

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