A007691 -id:A007691 - cp's OEIS frontend

A245775 Numbers k such that A017666(k) = denominator(sigma(k)/k) = 3.

3, 12, 84, 234, 270, 1080, 1488, 1638, 6048, 6552, 24384, 35640, 199584, 435708, 2142720, 4713984, 12999168, 18506880, 36197280, 100651008, 208565280, 240589440, 275890944, 299980800, 470564640, 3899750400, 4138364160, 6039429120, 13286744064, 17827568640

Offset: 1

Jaroslav Krizek, Aug 26 2014

nonn

Numbers n such that sigma(n)/n = k + 1/3 with integer k are terms of this sequence (3, 12, 234, 1080, 6048, 6552, 435708, 4713984, ...).
Subsequence of A245774 (numbers n such that n divides 3*sigma(n)).
Union of A160320 (sigma(n)/n = k + 1/3) and A160321 (sigma(n)/n = k + 2/3). - Michel Marcus, Aug 27 2014

			Number 12 is in sequence because A017666(12) = 3.

Jud McCranie, Table of n, a(n) for n = 1..44 (terms <= 10^13).

Cf. A000203, A007691, A160320, A160321, A245774.

[n: n in [1..3000000] | Denominator((SumOfDivisors(n))/n) eq 3]

for(n=1,10^7,if(denominator(sigma(n)/n)==3,print1(n,", "))) \\ Derek Orr, Aug 26 2014

More terms from A160320 and A160321 by Michel Marcus, Aug 27 2014

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

A259307 Numbers that belong to at least one amicable multiset.

Original entry on oeis.org

1, 6, 28, 120, 220, 284, 496, 672, 1184, 1210, 1560, 1740, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 7380, 7776, 8128, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 19260, 20448, 20640, 20664, 21168, 21384, 21924, 22200, 22428, 22752

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

Call a finite multiset {x_1, x_2, ..., x_k} of natural numbers (the x_i need not be distinct) an amicable multiset iff sigma(x_1)=sigma(x_2)=...=sigma(x_k)=x_1+x_2+...+x_k.
By definition, A255215 is a subset because a set can be regarded as a special multiset.
Also A007691 is a subset, since a k-perfect number corresponds to an amicable multiset in an obvious way. For example, since 120 is 3-perfect, the multiset {120, 120, 120} is amicable.
The first amicable multiset that belongs to neither A255215 nor A007691 is {1740, 1740, 1560}.

Jeppe Stig Nielsen, List of all amicable multisets with a sigma value below 10^7.

Cf. A255215, A007691, A259302, A259303, A259304, A259305, A259306.

/* write amicable multisets */ sMax=10^7;sigmaVals=vector(sMax,x,[]);for(n=1,sMax,s=sigma(n);s<=sMax&sigmaVals[s]=concat(sigmaVals[s],[n]));(MultisetSum(numbers,desiredSum,track)=if(desiredSum<0,return);if(desiredSum==0,print(apply(x->numbers[x],track));return);for(i=if(track,track[#track],1),#numbers,MultisetSum(numbers,desiredSum-numbers[i],concat(track,[i]))));for(s=1,sMax,MultisetSum(sigmaVals[s],s,[]))

A046763 Numbers that divide the sum of cubes of their divisors.

Original entry on oeis.org

1, 6, 42, 120, 168, 270, 280, 312, 496, 672, 728, 840, 1080, 1560, 1782, 1806, 1890, 2044, 2184, 2520, 3472, 3640, 3913, 4256, 5880, 6048, 6552, 6615, 7224, 7560, 7826, 8128, 9120, 9424, 9933, 10804, 10920, 11400, 12040, 12768, 13230, 13626, 14040

Offset: 1

Labos Elemer

nonn

Compare with multiply perfect numbers, A007691. Here Sum(divisors) is replaced by Sum(cube of divisors).
Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006
Tomohiro Yamada found that the odd number 209195 is a term. (See the Editorial Comment after the solution to Problem 11090.) - Jonathan Sondow, Nov 23 2012

			For k = 168 = a(5), Sum(d^3) = 5634720 = 33540*168 = 33540*k;

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A001158, A007691.

Select[Range[10^4], Divisible[DivisorSigma[3, #], #] &] (* Amiram Eldar, Sep 10 2019 *)

is(n)=sigma(n,3)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A093034 Primes of the form 1 + multiple perfect number.

Original entry on oeis.org

2, 7, 29, 673, 30241, 523777, 2178541, 23569921, 33550337, 66433720321, 137438691329, 30823866178561, 796928461056001, 1802582780370364661761, 9186050031556349952001, 2827987212986831882236723201, 630532357710420079508428362350593, 1928622300236318049928258133164033

Offset: 1

Labos Elemer, May 12 2004

nonn

From M. F. Hasler and Farideh Firoozbakht, Apr 08 2010: (Start)
Theorem: If p is a term of this sequence and p-1 is a t-perfect number then for each positive integer k, x=p^k is a solution to the equation sigma(phi(x)) = t*(x-1).
Proof: sigma(phi(x)) = sigma(phi(p^k)) = sigma((p-1)*p^(k-1)) = sigma(p-1)*sigma(p^(k-1)) = t*(p-1)*(p^k-1)/(p-1) = t*(p^k-1) = t*(x-1). (End)

Amiram Eldar, Table of n, a(n) for n = 1..35

Cf. A000010 (phi), A000203 (sigma), A007691, A171263, A171264, A171265.

a(11)-a(16) from Donovan Johnson, Nov 30 2008

a(17)-a(18) from Amiram Eldar, May 09 2024

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).

Original entry on oeis.org

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

A325637 Numbers k for which gcd(2k, sigma(k)) = 2k.

Original entry on oeis.org

6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128

Offset: 1

Antti Karttunen, May 21 2019

nonn

Multiply-perfect numbers (A007691) k with an even abundancy index sigma(k)/k. - Amiram Eldar, Jun 26 2024

Antti Karttunen, Table of n, a(n) for n = 1..1013 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)
Index entries for sequences related to sigma(n).
Index entries for sequences where any odd perfect numbers must occur.

Cf. A000203, A033879, A224832, A325636, A325638, A325654.
Subsequences: A000396, A336702 (after its initial 1).
Subsequence of A007691.

isA325637(n) = ((n+n)==gcd(n+n,sigma(n)));

a(n) = A224832(n)/2. - Amiram Eldar, Jun 26 2024

A326194 Number of iterations of x -> A009194(x) needed to reach a fixed point when starting from x = n, where A009194(x) = gcd(x, sigma(x)).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

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

Cf. A000203, A007691 (positions of zeros), A009194, A326195, A326196.

A326194(n) = { my(u=gcd(n,sigma(n))); if(u==n,0,1+A326194(u)); };

If gcd(n,sigma(n)) = n, then a(n) = 0, otherwise a(n) = 1 + a(gcd(n,sigma(n))).

a(n) < A326196(n).

A349685 Irregular triangle read by rows: the n-th row contains the elements in the continued fraction of the abundancy index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 3, 1, 5, 2, 1, 7, 1, 1, 7, 1, 2, 4, 1, 1, 4, 1, 11, 2, 3, 1, 13, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 15, 1, 17, 2, 6, 1, 19, 2, 10, 1, 1, 1, 10, 1, 1, 1, 1, 3, 1, 23, 2, 2, 1, 4, 6, 1, 1, 1, 1, 1, 2, 1, 2, 13, 2, 1, 29, 2, 2, 2, 1, 31, 1, 1, 31

Offset: 1

Amiram Eldar, Nov 25 2021

The abundancy index of n is sigma(n)/n = A000203(n)/n = A017665(n)/A017666(n).
For a prime p, the p-th row has a length 2 with a(p, 1) = 1 and a(p, 2) = p.
For multiply-perfect numbers m (A007691), the m-th row has a length 1, since their abundancy index is an integer. In particular, for a perfect number m (A000396), the m-th row has a length 1 with a(m, 1) = 2.

			The first ten rows of the triangle are:
1,
1, 2,
1, 3,
1, 1, 3,
1, 5,
2,
1, 7,
1, 1, 7,
1, 2, 4,
1, 1, 4,
...

Amiram Eldar, Table of n, a(n) for n = 1..10489 (rows 1..2000)

Cf. A000203, A000396, A007691, A017665, A017666, A071862 (row lengths), A071865, A349473.

row[n_] := ContinuedFraction[DivisorSigma[1, n]/n]; Table[row[k], {k, 1, 32}] // Flatten

row(n) = contfrac(sigma(n)/n); \\ Michel Marcus, Nov 25 2021

A379486 Numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 4, 6, 14, 16, 18, 24, 26, 28, 40, 54, 62, 64, 66, 74, 86, 102, 114, 122, 134, 138, 146, 152, 162, 169, 174, 176, 182, 184, 186, 206, 222, 234, 254, 270, 280, 282, 289, 290, 302, 304, 306, 308, 314, 318, 326, 338, 342, 354, 360, 361, 366, 368, 380, 384, 386, 402, 414, 422, 426, 434, 438, 441, 446, 448, 456, 474, 496

Offset: 1

Antti Karttunen, Jan 01 2025

nonn

Cf. A000203, A003961, A276086, A322361, A324198, A324644, A342671, A379485 (characteristic function), A379487, A379488.
Positions of 0's in A379489.
Cf. A379491 (subsequence, terms that are multiperfect numbers, A007691).

PARI
```
is_A379486 = A379485;
```

{Numbers k such that A379487(k) = A379488(k)}.

{Numbers k such that A322361(k)/A324198(k) = A324644(k)/A342671(k)}.

cp's OEIS Frontend

A245775 Numbers k such that A017666(k) = denominator(sigma(k)/k) = 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

Extensions

A245782 Refactorable multiply-perfect numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Extensions

A259307 Numbers that belong to at least one amicable multiset.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A046763 Numbers that divide the sum of cubes of their divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A093034 Primes of the form 1 + multiple perfect number.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A325637 Numbers k for which gcd(2k, sigma(k)) = 2k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A326194 Number of iterations of x -> A009194(x) needed to reach a fixed point when starting from x = n, where A009194(x) = gcd(x, sigma(x)).

Views

A379486 Numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.