A356191 -id:A356191 - cp's OEIS frontend

A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356193 at n=16.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A013664, A268335, A335988.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};

Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A335988.
a(n) = A356191(n) iff n is a powerful number (A001694).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022

A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 28 2020

a(n) is the least number k such that k*n (and also n/k) is an exponentially odd number (A268335). - Amiram Eldar, Nov 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A007913, A007947, A059841, A268335, A336644, A350390, A356191.

f[p_, e_] := p^(1 - Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336643(n) = (factorback(factorint(n)[, 1]) / core(n));

A336643(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(1-(f[i, 2]%2))));

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X^2) * (1 + X + p*X^2 - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336643(n): return prod(primefactors(n))//core(n) # Chai Wah Wu, Dec 30 2021

def A336643(n: int) -> int:
    return prod(b^(1 - e % 2) for (b, e) in list(factor(n)))
print([A336643(n) for n in range(1, 106)])  # Peter Luschny, Aug 23 2025

a(n) = A007947(n) / A007913(n).
Multiplicative with a(p^k) = p^(1-(k mod 2)) = p^A059841(k).
a(n) = n/A350390(n). - Amiram Eldar, Jan 01 2022
a(n) = A356191(n)/n. - Amiram Eldar, Nov 18 2022
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Sep 09 2023
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(1-5*s) + p^(2-5*s) + 2*p^(1-4*s) - p^(2-4*s) - p^(1-3*s) + p^(-3*s) - 2*p^(-2*s)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 12 * (log(n) + 3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.217778716619536378323007514119446813130797755001355937648276403523626491...,
f'(1) = f(1) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.716596820856763078660955244870812634072512131626849517007098664560806248...
and gamma is the Euler-Mascheroni constant A001620. (End)

A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 16, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356192 at n=16.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A036966, A360541.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356194.

f[p_, e_] := p^Max[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^max(f[i,2],3))};

Multiplicative with a(p^e) = p^max(e,3).
a(n) = n iff n is in A036966.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p-2)/(p^3*(p-1))) = 1.76434793373691907811... . - Amiram Eldar, Jul 29 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(3)/4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.1559111567... . - Amiram Eldar, Nov 13 2022
a(n) = n * A360541(n). - Amiram Eldar, Sep 01 2023

A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67

Offset: 1

Amiram Eldar, Jul 29 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A138302, A353897.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193.

f[p_, e_] := p^(2^Ceiling[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(e=logint(n,2)); if(n == 2^e, n, 2^(e+1))};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^s(f[i,2]))};

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))).

a(n) = n iff n is in A138302.

A011264 In the prime factorization of n, increment even powers and decrement odd powers (multiplicative).

Original entry on oeis.org

1, 1, 1, 8, 1, 1, 1, 4, 27, 1, 1, 8, 1, 1, 1, 32, 1, 27, 1, 8, 1, 1, 1, 4, 125, 1, 9, 8, 1, 1, 1, 16, 1, 1, 1, 216, 1, 1, 1, 4, 1, 1, 1, 8, 27, 1, 1, 32, 343, 125, 1, 8, 1, 9, 1, 4, 1, 1, 1, 8, 1, 1, 27, 128, 1, 1, 1, 8, 1, 1, 1, 108, 1, 1, 125, 8, 1, 1, 1, 32, 243, 1, 1, 8, 1, 1, 1, 4, 1, 27, 1, 8, 1, 1

Offset: 1

Marc LeBrun

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A001221, A004442, A007913, A007947, A011262, A027748, A336643, A356191.

a011264 n = product $ zipWith (^)
                      (a027748_row n) (map a004442 $ a124010_row n)
-- Reinhard Zumkeller, Jun 23 2013

f[n_, k_] := n^(If[EvenQ[k], k + 1, k - 1]); Table[Times @@ f @@@ FactorInteger[n], {n, 94}] (* Jayanta Basu, Aug 14 2013 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i,2]%2, f[i,2]-1, f[i,2]+1));} \\ Amiram Eldar, Jan 07 2023

a(n) = Product_{k=1..A001221(n)} (A027748(n,k)^A004442(A124010(n,k))). - Reinhard Zumkeller, Jun 23 2013
From Amiram Eldar, Jan 07 2023: (Start)
a(n) = n^2/A011262(n).
a(n) = n*A007947(n)/A007913(n)^2.
a(n) = n*A336643(n)/A007913(n).
a(n) = A356191(n)/A007913(n). (End)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-3) - 1/p^(2*s-2)). - Amiram Eldar, Sep 21 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Dirichlet g.f.: zeta(2*s-3) * Product_{p prime} (1 + (p-1)*p^(3-2*s) + p^(1-s) - (p-1)*(p^s + p^3)/(p^(2*s) - p^2)).
Sum_{k=1..n} a(k) ~ n^2/4. (End)

A355038 a(n) = n^2 times the squarefree kernel of n.

Original entry on oeis.org

1, 8, 27, 32, 125, 216, 343, 128, 243, 1000, 1331, 864, 2197, 2744, 3375, 512, 4913, 1944, 6859, 4000, 9261, 10648, 12167, 3456, 3125, 17576, 2187, 10976, 24389, 27000, 29791, 2048, 35937, 39304, 42875, 7776, 50653, 54872, 59319, 16000, 68921, 74088, 79507, 42592, 30375

Offset: 1

Peter Munn, Jun 16 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

The range of values is A335988.

Cf. A007947, A064549, A065463, A102631, A356191.

a[n_] := n^2 * Times @@ FactorInteger[n][[;; , 1]]; Array[a, 50] (* Amiram Eldar, Jun 18 2022 *)

PARI
```
a(n) = n^2 * factorback(factor(n)[,1]);
```

Multiplicative with a(p^e) = p^(2e+1).
a(n) = n^2 * A007947(n).
a(n) = A064549(n^2). - Amiram Eldar, Jun 20 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 - 1/(p*(p+1))) = A065463 / 4 = 0.1761105502... . - Amiram Eldar, Nov 13 2022
a(n) = A356191(n^2). - Amiram Eldar, Nov 30 2023

A365349 The sum of divisors of the smallest exponentially odd number divisible by n.

Original entry on oeis.org

1, 3, 4, 15, 6, 12, 8, 15, 40, 18, 12, 60, 14, 24, 24, 63, 18, 120, 20, 90, 32, 36, 24, 60, 156, 42, 40, 120, 30, 72, 32, 63, 48, 54, 48, 600, 38, 60, 56, 90, 42, 96, 44, 180, 240, 72, 48, 252, 400, 468, 72, 210, 54, 120, 72, 120, 80, 90, 60, 360, 62, 96, 320

Offset: 1

Amiram Eldar, Sep 02 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000203, A356191, A365348.

f[p_, e_] := (p^(e + 2 - Mod[e, 2]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + 2 - f[i,2]%2) - 1)/(f[i,1] - 1));}

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^3*X^3) )[n], ", ")) \\ Vaclav Kotesovec, Sep 04 2023

a(n) = A000203(A356191(n)).
Multiplicative with a(p^e) = (p^(e + 2 - (e mod 2)) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-3) - 1/p^(3*s-3)).
From Vaclav Kotesovec, Sep 04 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s-2) * zeta(2*s-3) * Product_{p prime} (1 - 1/p^(6*s-7) + 1/p^(5*s-6) + 1/p^(5*s-7) + 1/p^(4*s-4) + 1/p^(4*s-5) - 1/p^(4*s-6) - 1/p^(3*s-3) - 1/p^(3*s-4) - 1/p^(2*s-2)).
Let f(s) = Product_{p prime} (1 - 1/p^(6*s-7) + 1/p^(5*s-6) + 1/p^(5*s-7) + 1/p^(4*s-4) + 1/p^(4*s-5) - 1/p^(4*s-6) - 1/p^(3*s-3) - 1/p^(3*s-4) - 1/p^(2*s-2)), then
Sum_{k=1..n} a(k) ~ n^2 * Pi^4 * f(2) / 144 * (log(n) + 3*gamma - 1/2 + 18*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 1/p^2) * (1 - 2/p^2 + 1/p^3) = 6*A065464/Pi^2 = 0.26034448085669554670553581687050222309091096557569931376863612821007515...,
f'(2) = f(2) * Sum_{p prime} 3*(3*p-2) * log(p) / (p^3 - 2*p + 1) = f(2) * 4.40861022247384449961018198035049309399000439627743168713608947117149645... and gamma is the Euler-Mascheroni constant A001620. (End)

A365480 The sum of unitary divisors of the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 3, 4, 9, 6, 12, 8, 9, 28, 18, 12, 36, 14, 24, 24, 33, 18, 84, 20, 54, 32, 36, 24, 36, 126, 42, 28, 72, 30, 72, 32, 33, 48, 54, 48, 252, 38, 60, 56, 54, 42, 96, 44, 108, 168, 72, 48, 132, 344, 378, 72, 126, 54, 84, 72, 72, 80, 90, 60, 216, 62, 96, 224, 129, 84

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest exponentially odd number that is divisible by n is the same as the number of unitary divisors of n, A034444(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A034444, A034448, A356191, A365349, A365479, A365481.

f[p_, e_] := p^(e + 1 - Mod[e, 2]) + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^(f[i,2] + 1 - f[i,2]%2) + 1);}

a(n) = A034448(A356191(n)).
Multiplicative with a(p^e) = p^(e + 1 - (e mod 2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-3) - 1/p^(2*s-2) - 1/p^(2*s-1) - 1/p^(3*s-3)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s-2) * zeta(2*s-3) * Product_{p prime} (1 - p^(7-6*s) - p^(5-5*s) + p^(7-5*s) + 2*p^(4-4*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(2-3*s) - p^(4-3*s) - p^(1-2*s) - 2*p^(2-2*s)).
Let f(s) = Product_{p prime} (1 - p^(7-6*s) - p^(5-5*s) + p^(7-5*s) + 2*p^(4-4*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(2-3*s) - p^(4-3*s) - p^(1-2*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ n^2 * Pi^4 * f(2) / 144 * (log(n) + 3*gamma - 1/2 + 18*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 2/p^3 + 3/p^4 - 2/p^5) = 0.17432153313226756485612314112586411632220602294650993976966957787608316...,
f'(2) = f(2) * Sum_{p prime} 11 * log(p) / (p^2 + p - 2) = f(2) * 5.12969275236278527949034734003948649118572887258486718244613616120875581...
and gamma is the Euler-Mascheroni constant A001620. (End)

A365348 The number of divisors of the smallest exponentially odd number divisible by n.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8, 2, 6, 4, 4, 4, 16, 2, 4, 4, 8, 2, 8, 2, 8, 8, 4, 2, 12, 4, 8, 4, 8, 2, 8, 4, 8, 4, 4, 2, 16, 2, 4, 8, 8, 4, 8, 2, 8, 4, 8, 2, 16, 2, 4, 8, 8, 4, 8, 2, 12, 6, 4, 2, 16, 4

Offset: 1

Amiram Eldar, Sep 02 2023

The sum of these divisors is A365349(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000005, A356191, A365349.

f[p_, e_] := e + 2 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> x + 2 - x%2, factor(n)[, 2]));

a(n) = A000005(A356191(n)).
Multiplicative with a(p^e) = e + 2 - (e mod 2).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Let f(s) = Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.74469549790606742043912387159445432811796913290492411186307181370150975026...
f'(1) = f(1) * Sum_{p prime} 2*(3*p - 2) * log(p) / (1 - 2*p + p^4) = f(1) * 0.75575434641494973924789411019492794958528241212857430737760075121773728338...
and gamma is the Euler-Mascheroni constant A001620. (End)

A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Apr 28 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036537, A372328.
Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193, A356194.
Differs from A102631 at n = 8, 24, 27, 32, 40, 54, 56, 64, ... .

f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(e=logint(n + 1, 2)); if(n + 1 == 2^e, n, 2^(e+1) - 1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i, 1]^s(f[i, 2]))};

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - 1).
a(n) = n * A372328(n).
a(n) = n if and only if n is in A036537.
a(n) <= n^2, with equality if and only if n = 1.

cp's OEIS Frontend

A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

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A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

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A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

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A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

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A011264 In the prime factorization of n, increment even powers and decrement odd powers (multiplicative).

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A355038 a(n) = n^2 times the squarefree kernel of n.

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A365349 The sum of divisors of the smallest exponentially odd number divisible by n.

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