A327158 -id:A327158 - 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

A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

Original entry on oeis.org

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

N. J. A. Sloane

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005
It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015
Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - Amiram Eldar, Mar 05 2019
Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

			Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974
M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.
G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits
C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972
C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number

Cf. A034460, A034448, A057447.
Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
Gives the positions of ones in A327159.

usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* Harvey P. Dale, Nov 14 2012 *)

is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 3, 19, 2, 21, 11, 23, 2, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 18, 37, 19, 39, 20, 41, 7, 43, 11, 3, 23, 47, 12, 49, 25, 17, 26, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of unitary divisors of n.

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A007947, A017666, A034448, A077610, A319677, A323166, A327158 (positions of 1's), A332881, A332882 (numerators).

a:= n-> denom(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Denominator

a(n) = denominator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = denominator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = denominator of usigma(n)/n.
a(p) = p, where p is prime.
a(n) = n / A323166(n). - Antti Karttunen, Nov 13 2021

A348601 Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).

Original entry on oeis.org

1, 6, 40, 234, 588, 89376, 10805558400

Offset: 1

Amiram Eldar, Oct 25 2021

The corresponding quotients A160135(k)/k are 1, 1, 1, 1, 1, 2, 3, ...

a(8) > 1.5*10^10, if it exists.

			6 is a term since its nonexponential divisors are 1, 2 and 3, so A160135(6) = 1 + 2 + 3 = 6 which is divisible by 6.
40 is a term since its nonexponential divisors are 1, 2, 4, 5, 8 and 20, so A160135(40) = 1 + 2 + 4 + 5 + 8 + 20 = 40 which is divisible by 40.

Cf. A160135.

Similar sequences: A007691, A064594, A064595, A189000, A327158.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[1000], Divisible[DivisorSigma[1, #] - esigma[#], #] &]

A327164 Number of iterations of x -> gcd(usigma(x),x) needed to reach a fixed point, where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A323166, A327158 (positions of zeros).

Cf. also A326194.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323166(n) = gcd(n, A034448(n));
A327164(n) = { my(u=A323166(n)); if(u==n,0,1+A327164(u)); }

A327163 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = gcd(n,usigma(n)) * (-1)^[gcd(n,usigma(n))==n], and usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 5, 2, 4, 6, 2, 2, 7, 2, 8, 2, 4, 2, 9, 2, 4, 2, 5, 2, 7, 2, 2, 6, 4, 2, 4, 2, 4, 2, 4, 2, 7, 2, 5, 10, 4, 2, 5, 2, 4, 6, 4, 2, 7, 2, 11, 2, 4, 2, 12, 2, 4, 2, 2, 2, 7, 2, 4, 6, 4, 2, 13, 2, 4, 2, 5, 2, 7, 2, 4, 2, 4, 2, 5, 2, 4, 6, 5, 2, 14, 15, 5, 2, 4, 16, 9, 2, 4, 6, 8, 2, 7, 2, 4, 6

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = -A323166(n) = -n when n is one of unitary multiply-perfect numbers (A327158), otherwise f(n) = A323166(n) = gcd(n,A034448(n))
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j) => A327164(i) = A327164(j).

Antti Karttunen, Table of n, a(n) for n = 1..87360

Cf. A034448, A305800, A323166, A327158, A327164.

Cf. also A326193.

up_to = 87360;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323166(n) = gcd(n, A034448(n));
Aux327163(n) = { my(u=A323166(n)); u*((-1)^(u==n)); };
v327163 = rgs_transform(vector(up_to, n, Aux327163(n)));
A327163(n) = v327163[n];

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

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A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

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A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

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A348601 Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).

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A327164 Number of iterations of x -> gcd(usigma(x),x) needed to reach a fixed point, where usigma is the sum of unitary divisors of n (A034448).

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A327163 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = gcd(n,usigma(n)) * (-1)^[gcd(n,usigma(n))==n], and usigma is the sum of unitary divisors of n (A034448).

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A332478 Number that are unitary norm-multiply-perfect numbers in Gaussian integers.

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A348584 Numbers k such that k | A328258(k).

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