A145551 -id:A145551 - cp's OEIS frontend

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

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

A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

Original entry on oeis.org

1, 459818240, 51001180160, 13188979363639752997731839211623940096, 5157152737616023231698245840143799191339008, 54530444405217553992377326508106948362108928, 133821156044600922812153118065015159487725568, 4989680372093758991515359988337845750507257510078971904

Offset: 1

Jaroslav Krizek, Jan 21 2019

nonn

Multiperfect numbers m such that sigma(m) divides sigma(sigma(m)).
Also k-multiperfect numbers m such that k*m is also multiperfect.
Corresponding values of numbers k(n) = sigma(a(n)) / a(n): 1, 3, 3, 5, 5, 5, 5, 5, ...
Corresponding values of numbers h(n) = sigma(k(n) * a(n)) / (k(n) * a(n)): 1, 4, 4, 6, 6, 6, 6, 6, ...
Number of k-multiperfect numbers m such that sigma(n) is also multiperfect for k = 3..6: 2, 0, 20, 0.
From Antti Karttunen, Mar 20 2021, Feb 18 2022: (Start)
Conjecture 1 (a): This sequence consists of those m for which sigma(m)/m is an integer (thus a term of A007691), and coprime with m. Or expressed in a slightly weaker form (b): {1} followed by those m for which sigma(m)/m is an integer, but not a divisor of m. In a slightly stronger form (c): For m > 1, sigma(m)/m is always the least prime not dividing m. This would imply both (a) and (b) forms.
Conjecture 2: This sequence is finite.
Conjecture 3: This sequence is the intersection of A007691 and A351458.
Conjecture 4: This is a subsequence of A349745, thus also of A351551 and of A351554.
Note that if there existed an odd perfect number x that were not a multiple of 3, then both x and 2*x would be terms in this sequence, as then we would have: sigma(x)/x = 2, sigma(2*x)/(2*x) = 3, sigma(6*x)/(6*x) = 4. See also the diagram in A347392 and A353365.
(End)
From Antti Karttunen, May 16 2022: (Start)
Apparently for all n > 1, A336546(a(n)) = 0. [At least for n=2..23], while A353633(a(n)) = 1, for n=1..23.
The terms a(1) .. a(23) are only cases present among the 5721 known and claimed multiperfect numbers with abundancy <> 2, as published 03 January 2022 under Flammenkamp's site, that satisfy the condition for inclusion in this sequence.
They are also the only 23 cases among that data such that gcd(n, sigma(n)/n) = 1, or in other words, for which the n and its abundancy are relatively prime, with abundancy in all cases being the least prime that does not divide n, A053669(n), which is a sufficient condition for inclusion in A351458.
(End)

			3-multiperfect number 459818240 is a term because number 3*459818240 = 1379454720 is a 4-multiperfect number.

Antti Karttunen, Table of n, a(n) for n = 1..23 (prepared from 1600 term b-file given in A007691).
Achim Flammenkamp, The Multiply Perfect Numbers Page
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences related to sigma(n)

Cf. A000396, A005820, A027687, A046060, A046061, A053669, A054030, A145551, A323652, A336546, A342659, A347392, A349745, A351458, A351551, A351554, A353365.

Intersection of any two of these three sequences: A007691, A019278, A066961.

[n: n in [1..10^6] | SumOfDivisors(n) mod n eq 0 and SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0];

ismulti(n) = (sigma(n) % n) == 0;
isok(n) = ismulti(n) && ismulti(sigma(n)); \\ Michel Marcus, Jan 26 2019

A187680 a(n) = (product of divisors of n) mod (sum of divisors of n).

Original entry on oeis.org

0, 2, 3, 1, 5, 0, 7, 4, 1, 10, 11, 20, 13, 4, 9, 1, 17, 21, 19, 20, 25, 16, 23, 36, 1, 4, 9, 0, 29, 0, 31, 8, 33, 22, 25, 83, 37, 4, 9, 40, 41, 48, 43, 8, 21, 28, 47, 88, 1, 8, 9, 76, 53, 96, 1, 16, 49, 34, 59, 120, 61, 4, 31, 1

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2011

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A007955, A145551, A188061.

A187680 := proc(n) A007955(n) mod numtheory[sigma](n) ; end proc:
seq(A187680(n),n=1..120) ; # R. J. Mathar, Mar 17 2011

Table[Mod[Times@@Divisors[n],DivisorSigma[1,n]],{n,70}] (* Harvey P. Dale, May 23 2021 *)
a[n_] := Mod[n^(DivisorSigma[0, n]/2), DivisorSigma[1, n]]; Array[a, 60] (* Amiram Eldar, Jun 18 2022 *)

a(n) = my(d=divisors(n)); vecprod(d) % vecsum(d); \\ Michel Marcus, Jan 29 2019

a(n) = A007955(n) mod A000203(n).

a(n) = 0 iff n is in A145551 and a(n) = 1 iff n is in A188061. - Amiram Eldar, Jun 18 2022

A324529 a(n) = lcm(sigma(n), pod(n)) where sigma(k) = the sum of divisors of k (A000203) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 6, 12, 56, 30, 36, 56, 960, 351, 900, 132, 12096, 182, 1176, 1800, 31744, 306, 75816, 380, 168000, 14112, 4356, 552, 1658880, 3875, 14196, 29160, 21952, 870, 810000, 992, 2064384, 17424, 31212, 58800, 917070336, 1406, 21660, 85176, 23040000, 1722, 6223392

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

			For n=4: a(4) = lcm(sigma(4), pod(4)) = lcm(7, 8) = 56.

Cf. A000005, A000203, A014567, A145551.

[LCM(SumOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 1000]]

a(n) = my(d=divisors(n)); lcm(vecsum(d), vecprod(d)); \\ Michel Marcus, Mar 05 2019

a(n) = pod(n) for numbers n in A145551.

a(n) = sigma(n) * pod(n) for numbers n in A014567.

A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

Original entry on oeis.org

1, 6, 30, 66, 102, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, 672, 690, 714, 840, 870, 924, 930, 966, 1122, 1320, 1410, 1428, 1518, 1590, 1638, 1722, 1770, 1890, 1932, 2130, 2226, 2280, 2310, 2346, 2370, 2670, 2730, 2760, 2838, 2970, 2982, 3102, 3162, 3210, 3360, 3444, 3486, 3498, 3570, 3720, 3780, 3948, 3990

Offset: 1

Ilya Gutkovskiy, Oct 19 2016

nonn

Intersection of A003601, A120736 and A145551.
Numbers k such that A000005(k)|A007955(k), A000203(k)|A007955(k) and A000005(k)| A000203(k).
Numbers k such that A000005(k)|A062981(k), A072861(k)|A062758(k) and A245656(k) = 1.

			a(2) = 6 because 6 has 4 divisors {1,2,3,6}, 1*2*3*6/4 = 9, 1*2*3*6/(1 + 2 + 3 + 6) = 3 and (1 + 2 + 3 + 6)/4 = 3 are integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..20000
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Divisor Function.
Wikipedia, Arithmetic number.

Cf. A000005, A000203, A003601, A007955, A062758, A062981, A072861, A120736, A145551, A245656.

with(numtheory): P:=proc(q) local a,b,k,n;for n from 1 to q do
a:=divisors(n); b:=mul(a[k],k=1..nops(a));
if type(sigma(n)/tau(n),integer) and type(b/sigma(n),integer) and
type(b/tau(n),integer) then print(n); fi;
od; end: P(10^5); # Paolo P. Lava, Oct 20 2016

Select[Range[4000], Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[1, #1]] && Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[0, #1]] && Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] & ]

A244668 Numerators of (product of divisors of n / sum of divisors of n).

Original entry on oeis.org

1, 2, 3, 8, 5, 3, 7, 64, 27, 50, 11, 432, 13, 49, 75, 1024, 17, 1944, 19, 4000, 441, 121, 23, 27648, 125, 338, 729, 392, 29, 11250, 31, 32768, 363, 578, 1225, 10077696, 37, 361, 1521, 256000, 41, 64827, 43, 21296, 30375, 529, 47, 63700992, 343, 125000, 289, 70304, 53, 354294, 3025

Offset: 1

Jaroslav Krizek, Jul 04 2014

			a(7) = 7 because the divisors of 7 are 1 and 7, so then (1 * 7) / (1 + 7) = 7 / 8.
a(8) = 64 because the divisors of 8 are 1, 2, 4 and 8, so then (1 * 2 * 4 * 8) / (1 + 2 + 4 + 8) = 64 / 15.
a(9) = 27 because the divisors of 9 are 1, 3 and 9, so then (1 * 3 * 9) / (1 + 3 + 9) = 27 / 13.

Cf. A007955, A000203, A145551, A244669, A244670.

For denominators see A244669.

[Numerator((&*[d: d in Divisors(n)]) / (&+[d: d in Divisors(n)])): n in [1..100]]

Table[Numerator[(Times@@Divisors[n])/(Plus@@Divisors[n])], {n, 50}] (* Alonso del Arte, Jul 05 2014 *)

a007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)) ;
a(n) = numerator(a007955(n)/sigma(n)); \\ Michel Marcus, Jul 05 2014

Numerators of (A007955(n) / A000203(n)).
a(n) = n if n = 1 or if n is prime.
a(n) < n if and only if n = 6.

A244669 Denominators of (product of divisors of n / sum of divisors of n).

Original entry on oeis.org

1, 3, 4, 7, 6, 1, 8, 15, 13, 9, 12, 7, 14, 6, 8, 31, 18, 13, 20, 21, 32, 9, 24, 5, 31, 21, 40, 1, 30, 1, 32, 63, 16, 27, 48, 91, 38, 15, 56, 9, 42, 2, 44, 21, 26, 18, 48, 31, 57, 93, 8, 49, 54, 5, 72, 15, 80, 45, 60, 7, 62, 24, 104, 127, 84, 1, 68, 63, 32, 9

Offset: 1

Jaroslav Krizek, Jul 04 2014

Denominators of (A007955(n) / A000203(n)).

For numerators see A244668.

			a(8) = 15 because A007955(8) / A000203(8) = 64 / 15.

Cf. A007955, A000203, A145551, A244668, A244670.

[Denominator((&*[d: d in Divisors(n)]) / (&+[d: d in Divisors(n)])): n in [1..1000]]

a007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)) ;
a(n) = denominator(a007955(n)/sigma(n)); \\ Michel Marcus, Jul 05 2014

a(n) = 1 for numbers from A145551 (numbers n such that product of divisors of n / sum of divisors of n is an integer).

A244670 Product of divisors of n / sum of divisors of n for n such that product of divisors of n is divisible by sum of divisors of n.

Original entry on oeis.org

1, 3, 392, 11250, 131769, 1568294784, 501126, 119439360000000, 6566468639062500, 39226324511250000, 15780962, 162778775259375000, 2966827112704, 22417760034702144000000000000, 1801703513076518898, 30261936128, 3531477204506250000, 5876813016539072244

Offset: 1

Jaroslav Krizek, Jul 04 2014

nonn

			For n = 4; A145551(4) = 30; A244668(30) = 11250.

Cf. A007955, A000203, A145551, A244668, A244669.

[Numerator((&*[d: d in Divisors(n)]) / (&+[d: d in Divisors(n)])): n in [1..10000] | Denominator((&*[d: d in Divisors(n)]) / (&+[d: d in Divisors(n)])) eq 1]

a(n) = A007955(A145551(n)) / A000203(A145551(n)) = A244668(A145551(n)).

A192853 Places n such that the two remainders A187680(n) and A191906(n) are both zero.

Original entry on oeis.org

6, 28, 120, 270, 496, 672, 924, 1320, 3948, 7980, 8128, 10920, 12690, 15456, 18018, 25296, 27930, 29190, 30240, 30294, 32760, 35640, 52080, 55692, 61770, 69936, 76986, 83160, 83580, 86814, 106950, 127218, 130200, 131040, 141360, 155610

Offset: 1

Juri-Stepan Gerasimov, Jul 11 2011

nonn

The even perfect numbers (A000396) are a subsequence.

Robert Israel, Table of n, a(n) for n = 1..214

Cf. A000396, A145551, A187680, A191906

filter:= proc(n) local Q,p,s;
  Q:= numtheory:-divisors(n) minus {n};
  p:= convert(Q,`*`); s:= convert(Q,`+`);
  p mod s = 0 and (p * n ) mod (s + n ) = 0
end proc:
select(filter, [$2..2*10^5]); # Robert Israel, Apr 22 2025

A247145 Composite numbers such that the product of the number's proper divisors is divisible by the sum of the number's proper divisors.

Original entry on oeis.org

6, 12, 24, 28, 40, 42, 56, 60, 90, 120, 140, 153, 216, 234, 270, 290, 360, 440, 496, 522, 568, 585, 588, 672, 708, 819, 924, 984, 992, 1001, 1170, 1316, 1320, 1365, 1431, 1780, 2016, 2184, 2295, 2296, 2299, 2464, 2466, 2655, 2832, 3100, 3344, 3420, 3627, 3724, 3948, 4320, 4336, 4416, 4680

Offset: 1

David Consiglio, Jr., Nov 20 2014

Equal to the indices of the zero terms that correspond to composite numbers in A191906.

			12 is on the list because the proper divisors of 12 are [1,2,3,4,6]. The product of these numbers is 144. Their sum is 16. 144 is divisible by 16.

Robert Israel, Table of n, a(n) for n = 1..1203

Cf. A145551.

filter:= proc(n)
       local d,p,s;
     if isprime(n) then return false fi;
     d:= numtheory:-divisors(n) minus {n};
     convert(d,`*`) mod convert(d,`+`) = 0;
end proc:
select(filter, [$2..10000]); # Robert Israel, Dec 16 2014

a247145[n_Integer] :=
Select[Select[Range[n], CompositeQ[#] &],
Divisible[Times @@ Most@Divisors[#], Plus @@ Most@Divisors[#]] &]; a247145[4680] (* Michael De Vlieger, Dec 15 2014 *)
fQ[n_Integer] := Block[{d = Most@Divisors@n}, Mod[Times @@ d, Plus @@ d] == 0]; Select[Range@4680, ! PrimeQ@# && fQ@# &] (* Michael De Vlieger, Dec 19 2014, suggested by Robert G. Wilson v *)

forcomposite(n=1,10^3,d=divisors(n);p=prod(i=1,#d-1,d[i]);if(!(p%(sigma(n)-n)),print1(n,", "))) \\ Derek Orr, Nov 27 2014

from functools import reduce
from operator import mul
def divs(n):
    for i in range(1, int(n / 2 + 1)):
        if n % i == 0:
            yield i
    yield n
g = []
for a in range(2, 100):
    q = list(divs(a))[0:-1]
    if reduce(mul, q, 1) % sum(q) == 0 and len(q) != 1:
        g.append(a)
print(g)

More terms from Derek Orr, Dec 03 2014

cp's OEIS Frontend

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

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A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

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A187680 a(n) = (product of divisors of n) mod (sum of divisors of n).

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A324529 a(n) = lcm(sigma(n), pod(n)) where sigma(k) = the sum of divisors of k (A000203) and pod(n) = the product of divisors of k (A007955).

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A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

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A244668 Numerators of (product of divisors of n / sum of divisors of n).

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PARI

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A244669 Denominators of (product of divisors of n / sum of divisors of n).

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PARI