A046985 -id:A046985 - cp's OEIS frontend

A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920, 156036748944739017459105792, 3638193973609385308194865152

Offset: 1

Labos Elemer

nonn

Colton proves that perfect numbers (A000396) cannot be refactorable.

			k = 45532800 is a term since s0 = d(k) = 384, s1 = sigma(k) = 571963392, and the four quotients s1/s0 = 474300, (k * s0)/s1 = 96, s1/k = 4 and k/s0 = 118580 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..305
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

Cf. A000005, A000203, A000396, A046985, A001599, A007691.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n * d, s] && Divisible[s, d] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !((k * d) % s) && !(s % d) && !(k % d);} \\ Amiram Eldar, May 09 2024

Let s1 = sigma(k) = A000203(k) be the sum of divisors of k and s0 = d(k) = A000005(k) be the number of divisors of k. Then, k is a term if s1/s0, (k * s0)/s1, s1/k, and k/s0 are all integers.

a(7)-a(13) from Donovan Johnson, Apr 09 2010

Edited and a(14)-a(16) added by Amiram Eldar, May 09 2024

A007691 Multiply-perfect numbers: n divides sigma(n).

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009
A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013
Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013
Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013
Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013
Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014
The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

			120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.

T. D. Noe, Table of n, a(n) for n=1..1600 (using Flammenkamp's data)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Anonymous, Multiply Perfect Numbers [broken link]
Eric Bach, Gary Miller, and Jeffrey Shallit, Sums of divisors perfect numbers and factoring, SIAM J. Comput. 15:4 (1986), pp. 1143-1154.
Robert D. Carmichael, A table of multiply perfect numbers, Bull. Amer. Math. Soc. 13 (1907), 383-386.
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.
Walter Nissen, Abundancy : Some Resources
Kaitlin Rafferty and Judy Holdener, On the form of perfect and multiperfect numbers, Pi Mu Epsilon Journal, Vol. 13, No. 5 (Fall 2011), pp. 291-298.
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See p. 11.
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Hyperperfect Number.
Index entries for sequences where any odd perfect numbers must occur

Complement is A054027. Cf. A000203, A054030.
Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.
Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.
Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n)))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].
Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012

Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)
Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
(* Third program: *)
Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)

for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))

from sympy import divisor_sigma as sigma
def ok(n): return sigma(n, 1)%n == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

More terms from Jud McCranie and then from David W. Wilson.

Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021

A007340 Numbers whose divisors' harmonic and arithmetic means are both integers.

Original entry on oeis.org

1, 6, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000

Offset: 1

N. J. A. Sloane

Intersection of A001599 and A003601.
The following are also in A046985: 1, 6, 672, 30240, 32760. Also contains multiply perfect (A007691) numbers. - Labos Elemer
The numbers whose average divisor is also a divisor. Ore's harmonic numbers A001599 without the numbers A046999. - Thomas Ordowski, Oct 26 2014, Apr 17 2022
Harmonic numbers k whose harmonic mean of divisors (A001600) is also a divisor of k. - Amiram Eldar, Apr 19 2022

			x = 270: Sigma(0, 270) = 16, Sigma(1, 270) = 720; average divisor a = 720/16 = 45 and integer 45 divides x, x/a = 270/45 = 6, but 270 is not in A007691.

G. L. Cohen, personal communication.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, Curious and interesting numbers, Penguin Books, p. 124.

Donovan Johnson, Table of n, a(n) for n = 1..847
G. L. Cohen, Email to N. J. A. Sloane, Apr. 1994
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.
Hisanori Mishima, Factorizations of many number sequences
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.

Intersection of A003601 and A001599.
Different from A090945.
Cf. A001600, A007691, A046985-A046987, A046999, A054025.

a007340 n = a007340_list !! (n-1)
a007340_list = filter ((== 0) . a054025) a001599_list
-- Reinhard Zumkeller, Dec 31 2013

filter:= proc(n)
uses numtheory;
local a;
a:= sigma(n)/sigma[0](n);
type(a,integer) and type(n/a,integer);
end proc:
select(filter, [$1..2500000]); # Robert Israel, Oct 26 2014

Do[ a = DivisorSigma[0, n]/ DivisorSigma[1, n]; If[IntegerQ[n*a] && IntegerQ[1/a], Print[n]], {n, 1, 2500000}] (* Labos Elemer *)
ahmQ[n_] := Module[{dn = Divisors[n]}, IntegerQ[Mean[dn]] && IntegerQ[HarmonicMean[dn]]]; Select[Range[2500000], ahmQ] (* Harvey P. Dale, Nov 16 2011 *)

is(n)=my(d=divisors(n),s=vecsum(d)); s%#d==0 && #d*n%s==0 \\ Charles R Greathouse IV, Feb 07 2017

a = Sigma(1, x)/Sigma(0, x) integer and b = x/a also.

More terms from Robert G. Wilson v, Oct 03 2002

Edited by N. J. A. Sloane, Oct 05 2008 at the suggestion of R. J. Mathar

A046987 Multiply perfect numbers that are neither harmonic numbers nor arithmetic numbers.

Original entry on oeis.org

120, 523776, 1476304896, 31998395520, 30823866178560, 69357059049509038080, 4010059765937523916800, 27099073228001299660800, 686498980761986918441287680, 2827987212986831882236723200, 115131961034430181728489308160, 13361233986454282110797768294400, 32789312424503984621373515366400

Offset: 1

Labos Elemer

nonn

			k = 523776 is a term since s0 = d(k) = 80, s1 = sigma(k) = 1571328, s1/k = 1571328/523776 = 3 is an integer, but (k * s0)/s1 = 80/3 and s1/s0 = 98208/5 are not integers.

Amiram Eldar, Table of n, a(n) for n = 1..100

In A007691 but neither in A003601 nor in A001599.

Cf. A000005, A000203, A046985, A046986.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && !Divisible[n * d, s] && !Divisible[s, d]]; Select[Range[6*10^5], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && ((k * d) % s) && (s % d);} \\ Amiram Eldar, May 09 2024

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, but (k * s0) is not divisible by s1, and s1 is not divisible by s0.

a(5)-a(10) from Donovan Johnson, Nov 30 2008

Edited and a(11)-a(13) added by Amiram Eldar, May 09 2024

A046986 Multiply perfect numbers that are also harmonic numbers but are not arithmetic numbers.

Original entry on oeis.org

28, 496, 8128, 2178540, 33550336, 142990848, 459818240, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128

Offset: 1

Labos Elemer

nonn

			k = 2178540 is a term since s0 = d(k) = 216 and s1 = sigma(k) = 8714160, s1/s0 = 8714160/216 = 121030/3 is not an integer, and (k * s0)/s1 = (2178540 * 216)/8714160 = 54 and s1/k = 8714160/2178540 = 4 are integers.

Amiram Eldar, Table of n, a(n) for n = 1..207

In A007691 and A001599 but not in A003601.

Cf. A000005, A000203, A046985, A046987.

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, s1 | (k * s0), but s1 is not divisible by s0.

a(12)-a(17) from Donovan Johnson, Nov 30 2008

Edited and a(18)-a(21) added by Amiram Eldar, May 09 2024

A046999 Numbers k whose average divisor is nonintegral and divides k.

Original entry on oeis.org

28, 496, 8128, 950976, 2178540, 33550336, 142990848, 301953024, 459818240, 675347400, 714954240, 995248800, 1379454720, 2701389600, 3288789504, 6720569856, 8589869056, 10200236032, 14254365440, 30600708096, 42763096320, 43861478400, 66433720320, 71271827200

Offset: 1

Labos Elemer

nonn

The sequence contains perfect numbers (A000396) and others. Most of them have only small prime factors.
The first three terms are in A007691 (multiply perfect numbers) but 950976 is not since sigma_1/k is not an integer.
sigma_0(k) is the number of divisors of k (A000005).
sigma_1(k) is the sum of the divisors of k [same as sigma(k)] (A000203).
Harmonic numbers that are not arithmetic numbers. Of the 937 harmonic numbers below 10^14 there are just 90 such terms, of them 13 are multiply perfect numbers. - Amiram Eldar, Jun 08 2020

			k=28, sigma_0=6, sigma_1=56, m=sigma_1/sigma_0=9.333... is not an integer, but k/m=3 is;
k=950976, m=2958592/84=3521.333... but k/m=27 is integral.

Amiram Eldar, Table of n, a(n) for n = 1..90 (terms below 10^14)

In A001599 but not in A003601.

Cf. A007691, A046985, A046986, A046987, A000396.

Average divisor = m = sigma_1(k)/sigma_0(k) is not an integer but k/m is.

More terms from Jud McCranie, Dec 25 2000

a(16)-a(24) from Donovan Johnson, Apr 22 2008

A047727 Average divisor is an integer (A003601) and the number is refactorable (A033950).

Original entry on oeis.org

1, 56, 60, 96, 132, 184, 204, 248, 276, 348, 376, 480, 492, 504, 564, 568, 612, 632, 636, 672, 708, 824, 852, 864, 996, 1016, 1056, 1068, 1208, 1212, 1248, 1284, 1336, 1356, 1520, 1528, 1572, 1592, 1632, 1644, 1656, 1784, 1788, 1824, 1908, 1912, 1980

Offset: 1

Labos Elemer

nonn

			x = 56, sigma(x) = 120, number of divisors of x = 8. 120/8 and 56/8 are integers.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A007691, A001599, A046985.

adiQ[n_]:=Module[{ds1=DivisorSigma[1,n],ds0=DivisorSigma[0,n]} ,Divisible[ ds1,ds0]&&Divisible[n,ds0]]; Select[Range[2000],adiQ] (* Harvey P. Dale, Apr 27 2012 *)

isok(n) = my(d = numdiv(n)); !(n % d) && !(sigma(n) % d); \\ Michel Marcus, Oct 15 2016

Both sigma_1(x)/sigma_0(x) and x/sigma_0(x) are integers. - clarified by Harvey P. Dale, Apr 27 2012

A330533 Multiply-perfect numbers m whose average divisor is not an integer.

Original entry on oeis.org

28, 120, 496, 8128, 523776, 2178540, 33550336, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 31998395520, 43861478400, 66433720320, 137438691328, 704575228896, 30823866178560, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904

Offset: 1

Jaroslav Krizek, Dec 17 2019

nonn

Multiply-perfect numbers m such that A(m) = sigma(m) / tau(m) is not an integer. Corresponding values of A(m): 28/3, 45/2, 496/5, 8128/7, 98208/5, 121030/3, 33550336/13, 5957952/5, ...

Union of A046986 and A046987. Complement of A046985 with respect to A007691.

Amiram Eldar, Table of n, a(n) for n = 1..307

Cf. A007691, A046985, A046986, A046987.

[m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(SumOfDivisors(m) / m)];

Select[Range[10^6], Divisible[(s = DivisorSigma[1, #]), #] && !Divisible[s, DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 19 2019 *)

A331724 Multiply-perfect numbers (A007691) that are arithmetic (A003601).

Original entry on oeis.org

1, 6, 672, 30240, 32760, 23569920, 45532800, 14182439040, 51001180160, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480

Offset: 1

Jaroslav Krizek, Jan 25 2020

nonn

Multiply-perfect numbers m such that values A(m) = sigma(m)/tau(m) = A000203(m)/A000005(m) are any integers.
Corresponding values of A(m): 1, 3, 84, 1260, 1365, 294624, 474300, 36933435, 318757376, 637514752, 1199497728, ...
Complement of A330533 with respect to A007691. Supersequence of A046985.
Has many terms in common with B = {multiply perfect numbers n divisible by bigomega(n)}: only {1, 45532800, 403031236608, 212517062615531520, ...} are in {a(n)} \ B, while {120, 523776, 2178540, ...} are in B \ {a(n)}. - M. F. Hasler, Jan 31 2020

			sigma(672)/tau(672) = 2016/24 = 84 (integers).

Amiram Eldar, Table of n, a(n) for n = 1..423

Cf. A000005, A000203, A003601, A007691, A046985, A330533.

Cf. A325025 (multiply-perfect numbers that are harmonic).

[m: m in [1..10^7] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(SumOfDivisors(m) / m)];

seqQ[n_] := And @@ (Divisible[DivisorSigma[1, n], #] & /@ {n, DivisorSigma[0, n]}); Select[Range[5*10^7], seqQ] (* Amiram Eldar, Jan 25 2020 *)

is_A331724(n)={my(f=factor(n),s=sigma(f));!(s%n||s%numdiv(f))} \\ M. F. Hasler, Jan 31 2020

cp's OEIS Frontend

A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

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A007691 Multiply-perfect numbers: n divides sigma(n).

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A007340 Numbers whose divisors' harmonic and arithmetic means are both integers.

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A046987 Multiply perfect numbers that are neither harmonic numbers nor arithmetic numbers.

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A046986 Multiply perfect numbers that are also harmonic numbers but are not arithmetic numbers.

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A046999 Numbers k whose average divisor is nonintegral and divides k.

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A047727 Average divisor is an integer (A003601) and the number is refactorable (A033950).

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