A325020 -id:A325020 - cp's OEIS frontend

A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

1, 6, 28, 496, 672, 8128, 30240, 32760, 332640, 695520, 2178540, 17428320, 23569920, 33550336, 45532800, 52141320, 142990848, 164989440, 318729600, 447828480, 481572000, 500860800, 540277920, 623397600, 644271264, 714954240, 995248800, 1047254400, 1307124000

Offset: 1

Jaroslav Krizek, Mar 27 2019

nonn

Numbers m such that m*tau(m)/sigma(m) is an integer g and simultaneously m*(m-tau(m))/sigma(m) is an integer h. Corresponding values of integers g: 1, 2, 3, 5, 8, 7, 24, 24, 44, 46, 54, 96, 80, 13, 96, ...
Corresponding values of integers h: 0, 1, 11, 243, 216, 4057, 7536, 8166, 76186, 166589, ...
Even perfect numbers from A000396 are terms.
Complement of A325022 with respect to A001599.
Intersection of A325020 and A001599.

			Harmonic number 28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..255 (terms below 10^14)

Cf. A000005, A000203, A001599, A007691, A049820, A325020, A325022, A325023, A325024.

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Select[Range[10^6], And[IntegerQ@ HarmonicMean@ #2, IntegerQ[#1 (#1 - #3)/#4]] & @@ Join[{#}, {Divisors@ #}, DivisorSigma[{0, 1}, #]] &] (* Michael De Vlieger, Mar 27 2019 *)

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && !frac(m*(m-d)/s); \\ Michel Marcus, Mar 27 2019

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325021_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not (n*n%s or reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s):
            yield n
A325021_list = list(islice(A325021_gen(),10)) # Chai Wah Wu, Feb 14 2023

A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 14182439040, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120, 181742883469056, 6088728021160320

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Numbers m such that sigma(m)/m is an integer f and simultaneously m*tau(m)/sigma(m) is an integer g. Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ...
Complement of A325024 with respect to A007691.
Even perfect numbers from A000396 are terms.
Intersection of A325020 and A007691.
Conjecture: Numbers m such that all values of sigma(m)/m, m*tau(m)/sigma(m) and m*(m-tau(m))/sigma(m) are any integers (f, g, and h respectively). Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, ...

			Multi-perfect number 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..512

Cf. A000005, A000203, A000396, A007691, A049820, A325020, A325021, A325022, A325024.

[n: n in [1..1000000] | IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[#3, #1] == 0, IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Mar 24 2019 *)

isok(m) = my(s=sigma(m)); (frac(m*(m-numdiv(m))/s) == 0) && (frac(s/m) == 0); \\ Michel Marcus, Mar 25 2019

A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 360360, 539400, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600, 4713984, 4754880, 5772200, 6051500

Offset: 1

Jaroslav Krizek, Mar 28 2019

nonn

Numbers m such that sigma(m) divides m*tau(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325021 with respect to A001599.

			140 is a term because 140*(140-tau(140))/sigma(140) = 140*(140-12)/336 = 160/3.

Amiram Eldar, Table of n, a(n) for n = 1..682 (terms below 10^14)

Cf. A000005, A000203, A001599, A049820, A325020, A325021, A325023, A325024.

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Select[Range[10^5], And[IntegerQ@ HarmonicMean@ #4, ! IntegerQ[#1 (#1 - #2)/#3]] & @@ Append[{#}~Join~DivisorSigma[{0, 1}, #], Divisors@ #] &] (* Michael De Vlieger, Mar 30 2019 *)

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && frac(m*(m-d)/s); \\ Michel Marcus, Mar 28 2019

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325022_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if n*n%s and not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
            yield n
A325022_list = list(islice(A325022_gen(),10)) # Chai Wah Wu, Feb 14 2023

A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

120, 523776, 459818240, 1476304896, 31998395520, 51001180160, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Jaroslav Krizek, May 12 2019

nonn

Numbers m such that m divides sigma(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325023 with respect to A007691.

			120 is a term because 120*(120-tau(120))/sigma(120) = 120*(120-16)/360 = 104/3.

Amiram Eldar, Table of n, a(n) for n = 1..218

Cf. A000005, A000203, A007691, A049820, A325020, A325021, A325022, A325023.

[n: n in [1..1000000] | not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[#3, #1] == 0, !IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Amiram Eldar, Jul 10 2019 after Michael De Vlieger at A325023 *)

isA325024(m) = { my(s=sigma(m)); ((1==denominator(s/m)) && (1!=denominator(m*(m-numdiv(m))/s))); }; \\ Antti Karttunen, May 25 2019

cp's OEIS Frontend

A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

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A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

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A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

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A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

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Examples

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Crossrefs

Programs

Magma

Mathematica

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