A054030 -id:A054030 - 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

A166069 Multiply perfect numbers k such that sigma(k)/k > 2.

Original entry on oeis.org

120, 672, 30240, 32760, 523776, 2178540, 23569920, 45532800, 142990848, 459818240, 1379454720, 1476304896, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, Oct 06 2009

nonn

Subsequence of multiply perfect numbers (A007691). Numbers k = A007691(n) such that sigma(A007691(n))/A007691(n) > 2. Numbers k = A007691(n) such that A054030(n) > 2.

			For n = 1 the a(1) = 120, sigma(120) / 120 = 360 / 120 = 3, i.e. > 2.

Charles R Greathouse IV, Table of n, a(n) for n=1..1587
Achim Flammenkamp, The Multiply Perfect Numbers Page

isok(n) = sn = sigma(n)/n ; (type(sn) == "t_INT") && (sn > 2); \\ Michel Marcus, Oct 24 2013

Extended by Charles R Greathouse IV, Oct 12 2009

A081756 Numbers n such that there is a proper divisor d of n satisfying sigma(d)=n.

Original entry on oeis.org

1, 12, 56, 360, 992, 2016, 16256, 120960, 131040, 1571328, 8714160, 67100672, 94279680, 182131200, 571963392, 1379454720, 4428914688, 5517818880, 17179738112, 70912195200, 153003540480, 159991977600, 175445913600, 265734881280, 274877382656, 612014161920

Offset: 1

Benoit Cloitre, Apr 08 2003

nonn

A139256 is a subsequence. - Michel Marcus, Dec 02 2013

Ray Chandler, Table of n, a(n) for n = 1..1000 (using comment in formula section from _David Wasserman_ and b-files for A007691 and A054030)

Cf. A007691, A054030, A065125, A139256.

kmax = 10^12;
A007691 = Cases[Import["https://oeis.org/A007691/b007691.txt", "Table"], {, }][[All, 2]];
A054030 = Cases[Import["https://oeis.org/A054030/b054030.txt", "Table"], {, }][[All, 2]];
okQ[n_] := AnyTrue[Most[Divisors[n]], DivisorSigma[1, #] == n&];
{1}~Join~Reap[Do[k = A007691[[i]]*A054030[[j]]; If[k <= kmax, Sow[k]], {i, Length[A007691]}, {j, Length[A054030]}]][[2, 1]] // Union // Select[#, okQ]& (* Jean-François Alcover, Oct 31 2019, after David Wasserman *)

Multiply A007691 by A054030 and sort the resulting sequence. - David Wasserman, Jun 28 2004

More terms from David Wasserman, Jun 28 2004

Description clarified by Ray Chandler, May 18 2017

A165701 Numbers n such that 5^n-6 is prime.

Original entry on oeis.org

2, 4, 5, 6, 10, 53, 76, 82, 88, 242, 247, 473, 586, 966, 1015, 1297, 1825, 2413, 2599, 2833, 5850, 5965, 6052, 27199, 49704, 79000

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Oct 30 2009

Numbers corresponding to the a(n) for n>11 are probable prime.
If Q is a 4-perfect number and gcd(Q, 5*(5^a(n)-6))=1 then m=5^(a(n)-1)
(5^a(n)-6)*Q is a solution of the equation sigma(x)=5(x+Q)(see comment lines of the sequence A058959). 142990848 is the smallest 4-perfect number m such that 5 doesn't divide m.
a(27) > 10^5. - Robert Price, Feb 03 2014

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1
H. Lifchitz, R. Lifchitz: PRP Top Records Search for 5^n-6.

Cf. A007691, A054030, A058959.

Mathematica
```
Do[If[PrimeQ[5^n-6],Print[n]],{n,8888}]
```

is(n)=ispseudoprime(5^n-6) \\ Charles R Greathouse IV, Jun 13 2017

a(24)-a(26) from Robert Price, Feb 03 2014

A132628 Numbers n for which the sum of their digits divides sigma(n).

Original entry on oeis.org

1, 6, 10, 11, 15, 19, 20, 22, 24, 30, 31, 33, 35, 42, 51, 52, 60, 62, 66, 68, 71, 79, 87, 90, 100, 101, 102, 103, 104, 105, 109, 110, 112, 114, 118, 120, 123, 124, 130, 132, 136, 138, 141, 143, 145, 147, 150, 153, 160, 161, 165, 167, 168, 174, 177, 190, 194, 198

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 24 2007

			n=147 -> sigma(n)=1+3+7+21+49+147=228 Sum_digits(n)=1+4+7=12 -> 228/12 = 19
n=177 -> sigma(n)=1+3+59+177=240 Sum_digits(n)=1+7+7=15 -> 240/15 = 16

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

Cf. A007691, A054030, A132629.

with(numtheory); P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; j:=sigma(i)/w; if trunc(j)=j then print(i,j); fi; od; end: P(200)

Select[Range[200],Divisible[DivisorSigma[1,#],Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Dec 06 2012 *)

A134639 Conjectured number of numbers k such that sigma(k)/k = n.

Original entry on oeis.org

6, 36, 65, 245, 516

Offset: 3

T. D. Noe, Nov 05 2007

These numbers come from Guy and Flammenkamp. Sequences A000396, A005820, A027687, A046060 and A046061 give the k for which the abundancy sigma(k)/k is 2, 3, 4, 5 and 6, respectively. Sequence A054030 gives the abundancy of each multiperfect number A007691.

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B2.

Achim Flammenkamp, The Multiply Perfect Numbers Page

A140798 Harmonic numbers that are not multiply-perfect.

Original entry on oeis.org

140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600

Offset: 1

Lekraj Beedassy, Jul 15 2008

nonn

Sequence is A001599 excluding those entries that appear in A007691.

Multiply-perfect numbers m (with sigma(m)/m an integer) are necessarily harmonic numbers (with tau(m)/{sigma(m)/m } an integer), but the converse is not true : If m divides sigma(m), then quotient sigma(m)/m divides tau(m) [m=A007691]; However, quotient tau(n)/{sigma(n)/n} being an integer does not imply quotient sigma(n)/n is necessarily an integer [n=A001599].

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 140, pp 48, Ellipses, Paris 2008.

Donovan Johnson, Table of n, a(n) for n = 1..912 (terms < 10^14)

Cf. A000203, A054030, A000005.

Select[Range[10^6], (d = Divisors[#]; IntegerQ[HarmonicMean[d]] && Mod[Total[d], #] != 0)&] (* Jean-François Alcover, Nov 13 2017 *)

Initial term 1 removed and offset corrected by Donovan Johnson, Jan 25 2012

A219545 Integer values of sigma(n)/n that are prime.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 5, 5, 3, 2, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 7, 2, 5, 5, 7, 5, 5, 5, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Jonathan Sondow, Nov 22 2012

nonn

Subsequence of A054030 consisting of primes among the abundancies sigma(m)/m of multiply perfect numbers m (see A007691).
Each 2 corresponds to a perfect number A000396, so if there are infinitely many perfect numbers, then the sequence is infinite.
If, in addition, there are only finitely many multiply perfect numbers m with sigma(m)/m > 2 (see A134639), then a(n) = 2 for all n > some N.

			A065997(1) = 6 and sigma(6)/6 = (1+2+3+6)/6 = 2, so a(1) = 2.

R. K. Guy, Unsolved Problems in Number Theory, B2.

A. Flammenkamp, The Multiply Perfect Numbers Page

Cf. A000396, A007691, A054030, A065997, A134639.

Select[Table[DivisorSigma[1,n]/n,{n,10^6}],PrimeQ] (* The program only generates the first seven terms of the sequence. To generate them all, the value of n would have to be greatly increased. *) (* Harvey P. Dale, Oct 25 2021 *)

a(n) = sigma(A065997(n))/A065997(n).

Extended by T. D. Noe, Nov 27 2012

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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A166069 Multiply perfect numbers k such that sigma(k)/k > 2.

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A081756 Numbers n such that there is a proper divisor d of n satisfying sigma(d)=n.

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A165701 Numbers n such that 5^n-6 is prime.

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A132628 Numbers n for which the sum of their digits divides sigma(n).

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A134639 Conjectured number of numbers k such that sigma(k)/k = n.

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A140798 Harmonic numbers that are not multiply-perfect.

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