A091507 -id:A091507 - cp's OEIS frontend

A093396 Denominators of n divided by the product of the anti-divisors of n.

2, 3, 6, 2, 30, 15, 4, 42, 42, 10, 270, 54, 8, 33, 2310, 280, 78, 78, 8, 4050, 4050, 14, 1428, 102, 440, 6270, 114, 32, 7938, 257985, 520, 138, 552, 16, 11250, 866250, 616, 1458, 1458, 2720, 14790, 174, 131040, 16926, 17670, 190, 39204, 78408, 8, 2315250

Offset: 3

Lior Manor, Mar 28 2004

See A066272 for definition of anti-divisor.

			The anti-divisors of 18 are 4, 5, 7, 12. Hence a(18) = 4*5*7*12/GCD(4*5*7*12, 18) = 280.

Dumitru Damian, Table of n, a(n) for n = 3..10000

Cf. A066417, A091507, A093394, A093395 (numerators).

import numpy as np
from sympy.ntheory.factor_ import antidivisors
def a093396(k):
        return (m:=np.prod(antidivisors(k), dtype=object))//np.gcd(m,k, dtype=object)
{print(a093396(k), end = ', ') for k in range(3,10**2)} # Dumitru Damian, Oct 16 2023

a(n) = A091507(n)/GCD(n, A091507(n))

Name changed by Franklin T. Adams-Watters, Aug 21 2013

A093394 a(n) is the GCD of n and the product of the anti-divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 15, 1, 1, 6, 1, 4, 21, 2, 1, 8, 5, 2, 27, 4, 1, 30, 1, 1, 33, 2, 35, 12, 1, 2, 39, 8, 1, 42, 1, 4, 45, 2, 1, 16, 7, 10, 51, 4, 1, 54, 55, 8, 57, 2, 1, 60, 1, 2, 63, 1, 65, 66, 1, 4, 69, 70, 1, 24, 1, 2, 75, 4

Offset: 3

Lior Manor, Mar 28 2004

nonn

See A066272 for definition of anti-divisor.

			The anti-divisors of 18 are 4, 5, 7, 12. Hence a(18) = gcd(4*5*7*12, 18) = 6.

Cf. A066417, A091507, A093395, A093396.

a(n) = gcd(n, A091507(n)).

A093395 Numerators of n divided by the product of the anti-divisors of n.

Original entry on oeis.org

3, 4, 5, 3, 7, 8, 3, 5, 11, 3, 13, 7, 1, 16, 17, 3, 19, 5, 1, 11, 23, 3, 5, 13, 1, 7, 29, 1, 31, 32, 1, 17, 1, 3, 37, 19, 1, 5, 41, 1, 43, 11, 1, 23, 47, 3, 7, 5, 1, 13, 53, 1, 1, 7, 1, 29, 59, 1, 61, 31, 1, 64, 1, 1, 67, 17, 1, 1, 71, 3, 73, 37, 1, 19

Offset: 3

Lior Manor, Mar 28 2004

See A066272 for definition of anti-divisor.

			The anti-divisors of 18 are 4, 5, 7, 12. Hence a(18) = 18/GCD(4*5*7*12, 18) = 3.

Cf. A066417, A091507, A093394, A093396 (denominators).

a(n) = n/GCD(n, A091507(n)) = n/A093394(n)

Name changed by Franklin T. Adams-Watters, Aug 21 2013

A130874 Anti-divisorial numbers: the product of all anti-divisors of all integers less than or equal to n.

Original entry on oeis.org

2, 6, 36, 144, 4320, 64800, 777600, 65318400, 2743372800, 109734912000, 29628426240000, 3199870033920000, 383984404070400000, 12671485334323200000, 29271131122286592000000, 49175500285441474560000000, 3835689022264435015680000000, 1196734974946503724892160000000

Offset: 3

Jonathan Vos Post, Jul 25 2007

nonn

Different from the anti-primorial, which is the partial products of anti-primes.

			a(11) = (anti-divisors of 3) * (anti-divisors of 4) * ... * (anti-divisors) of 11 = (2) * (3) * (2 * 3) * (4) * (2 * 3 * 5) * (3 * 5) * (2 * 6) * (3 * 4 * 7) * (2 * 3 * 7) = 2743372800.

Hakan Icoz, Table of n, a(n) for n = 3..150

Cf. A066272, A091507, A130799.

A130874 :=  proc(n)
    mul( A091507(i),i=1..n) ;
end proc:
seq(A130874(n),n=3..20) ; # R. J. Mathar, Jan 24 2022

from sympy.ntheory.factor_ import antidivisors
def A130874():
    sum = 1
    i = 2 #(offset-1)
    while True:
        i += 1
        for j in antidivisors(i):
            sum *= j
        yield sum
        if i == 50:#Generator stops after calculating a(50)
            break
for i in A130874():
    print(i) # Hakan Icoz, Dec 26 2021

a(n) = Product_{k=3..n} {anti-divisors(k)} = Product_{k=3..n} Product_{j=1..A066272(k)} (j-th element of k-th row of A130799) = partial products of A091507.

More terms from Hakan Icoz, Dec 25 2021

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A093396 Denominators of n divided by the product of the anti-divisors of n.

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A093394 a(n) is the GCD of n and the product of the anti-divisors of n.

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A093395 Numerators of n divided by the product of the anti-divisors of n.

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A130874 Anti-divisorial numbers: the product of all anti-divisors of all integers less than or equal to n.

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