A011775 -id:A011775 - 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

A015706 Odd numbers k that divide phi(k)*sigma(k).

Original entry on oeis.org

1, 117, 135, 775, 819, 891, 1521, 1701, 2325, 3159, 3375, 4455, 5733, 6875, 6975, 10935, 11907, 19773, 20625, 20925, 22113, 22275, 24025, 40131, 41067, 43875, 44375, 49005, 61875, 62775, 68607, 72075, 75625, 83349, 84375, 85293

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..3000
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A011775.

Select[Range[1,85301,2],Divisible[EulerPhi[#]DivisorSigma[1,#],#]&] (* Harvey P. Dale, Aug 27 2013 *)

isok(n) = (n%2) && !(sigma(n)*eulerphi(n) % n); \\ Michel Marcus, Oct 02 2017

A228104 Numbers of form 2^(2i-1)*3^j, with i,j > 0.

Original entry on oeis.org

6, 18, 24, 54, 72, 96, 162, 216, 288, 384, 486, 648, 864, 1152, 1458, 1536, 1944, 2592, 3456, 4374, 4608, 5832, 6144, 7776, 10368, 13122, 13824, 17496, 18432, 23328, 24576, 31104, 39366, 41472, 52488, 55296, 69984, 73728, 93312, 98304, 118098, 124416, 157464, 165888

Offset: 1

Ralf Stephan, Aug 10 2013

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A033845 and A011775.

N:= 10^6: # for terms <= N
sort([seq(seq(2^i * 3^j, j = 1 .. ilog[3](N/2^i)),i=1..ilog2(N/3),2)]); # Robert Israel, Oct 14 2024

With[{max = 2*10^5}, Flatten[Table[2^(2*i-1)*3^j, {i, 1, (Log2[max]+1)/2}, {j, 1, Log[3, max/2^(2*i-1)]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

vecsort(vector(10000,n,2^(2*((n-1)%100)+1)*3^((n\100)+1))) /* (first 100 values) */

Sum_{n>=1} 1/a(n) = 1/3. - Amiram Eldar, Mar 29 2025

A055196 Primitive numbers k that divide sigma(k)*phi(k).

Original entry on oeis.org

6, 28, 40, 117, 135, 496, 775, 891, 1701, 2176, 5632, 6875, 8128, 18688, 23552, 26624, 44375, 88723, 89667, 91136, 274625, 352256, 557375, 628849, 722701, 796797, 1071875, 1200663, 1288625, 1358127, 1512875, 2246503, 2473984, 2490368, 2896363, 2909375

Offset: 1

Robert G. Wilson v, Jun 30 2000

Terms from A011775 that are not a multiple of a previous term. - Michel Marcus, Dec 21 2013

Michel Marcus, Table of n, a(n) for n = 1..68

Cf. A011775.

Extended using A011775 b-file by Michel Marcus, Dec 21 2013

A066993 Integer of the form phi(n)*sigma(n)/n.

Original entry on oeis.org

1, 4, 13, 20, 24, 36, 40, 65, 64, 84, 112, 96, 128, 121, 171, 186, 200, 216, 168, 208, 192, 273, 312, 340, 392, 364, 480, 448, 456, 496, 612, 605, 576, 640, 768, 768, 840, 880, 840, 936, 960, 1105, 992, 1200, 1093, 1280, 1464, 1364, 1152, 1539, 1152, 1664, 1482

Offset: 1

Benoit Cloitre, Jan 27 2002

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A011775.

Select[Table[EulerPhi[n] DivisorSigma[1,n]/n,{n,2000}],IntegerQ] (* Harvey P. Dale, Mar 17 2020 *)

{ n=0; for (m=1, 10^10, if ((a=eulerphi(m)*sigma(m)/m) % 1.0 == 0 , write("b066993.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Apr 16 2010

a(n) = sigma(A011775(n))*phi(A011775(n))/A011775(n).

Missing term a(43)=992 and new term a(53)=1482 added by Harry J. Smith, Apr 16 2010

A066994 Numbers k such that phi(k) divides k*sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 24, 26, 27, 30, 32, 35, 36, 39, 40, 42, 48, 54, 55, 56, 60, 63, 64, 70, 72, 78, 80, 84, 88, 96, 98, 104, 105, 108, 110, 114, 116, 120, 125, 126, 128, 135, 140, 144, 147, 150, 155, 156, 160, 162, 165, 168, 189, 190, 192

Offset: 1

Benoit Cloitre, Jan 27 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010 (phi), A000203 (sigma), A011775.

Subsequences: A007694, A020492.

Select[Range[200],Divisible[# DivisorSigma[1,#],EulerPhi[#]]&] (* Harvey P. Dale, Aug 23 2019 *)

isok(k) = { (k*sigma(k)) % eulerphi(k) == 0 } \\ Harry J. Smith, Apr 23 2010

Missing term a(7)=10 added by Harry J. Smith, Apr 23 2010

A066995 Numbers k such that sigma(k) divides k*phi(k).

Original entry on oeis.org

1, 6, 15, 28, 84, 95, 140, 182, 190, 248, 270, 287, 308, 357, 420, 455, 477, 496, 570, 672, 744, 819, 840, 910, 1199, 1428, 1488, 1547, 1638, 1722, 1848, 1892, 2295, 2398, 2480, 2660, 2730, 3339, 3417, 3472, 3515, 3596, 3640, 3720, 3780, 3956, 4064, 4095

Offset: 1

Benoit Cloitre, Jan 27 2002

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001599, A011775.

Filtered([1..10^5], n ->  n*Phi(n) mod Sigma(n) = 0); # Muniru A Asiru, Jan 31 2018

Select[Range[4500],Divisible[# EulerPhi[#],DivisorSigma[1,#]]&]  (* Harvey P. Dale, Mar 19 2011 *)

isok(n) = frac(n*eulerphi(n)/sigma(n)) == 0; \\ Michel Marcus, Jan 31 2018

A067575 Numbers k that divide phi(k)*bigomega(k).

Original entry on oeis.org

1, 4, 12, 16, 18, 27, 64, 80, 96, 144, 200, 216, 256, 324, 448, 486, 500, 672, 729, 768, 1008, 1024, 1152, 1250, 1512, 1568, 1728, 2268, 2352, 2560, 2592, 3125, 3402, 3528, 3888, 4096, 5103, 5292, 5488, 5832, 6144, 6400, 7938, 8232

Offset: 1

Benoit Cloitre, Jan 30 2002

nonn

All powers of 4 (A000302) are in the sequence.

Cf. A000302, A011775.

Select[Range[8300],Mod[EulerPhi[#]PrimeOmega[#],#]==0&] (* Harvey P. Dale, Oct 23 2024 *)

A047630 Numbers k that divide sigma(k) * phi(k) and are not divisible by 6.

Original entry on oeis.org

1, 28, 40, 117, 135, 196, 200, 224, 496, 640, 775, 819, 891, 1000, 1372, 1521, 1550, 1568, 1701, 1792, 2176, 2325, 2480, 3100, 3159, 3200, 3375, 3724, 4455, 5000, 5632, 5733, 6200, 6860, 6875, 6975, 8128, 9604, 10240, 10880, 10935, 10976, 11907

Offset: 1

Robert G. Wilson v, Jul 22 2000

George E. Andrews, "Number Theory," Dover Books, NY, 1971, Page 84.

Cf. A011775.

Do[ If[ Mod[ n, 6 ]!=0, If[ Mod[ DivisorSigma[ 1, n ]*EulerPhi[ n ], n ]==0, Print[ n ] ] ], {n, 1, 25000} ]

A274205 Numbers such that the sum of divisors is twice the sum of the exponential divisors.

Original entry on oeis.org

6, 24, 54, 216, 1638, 6552, 14256, 55860, 80262, 276822, 321048, 502740, 1107288, 1396500, 1724976, 12568500, 13564278, 20165460, 54257112, 168836850, 181489140, 504136500, 675347400, 4537228500, 28533427650, 60950102850, 114133710600, 162252212850, 243800411400, 649008851400, 734916514878

Offset: 1

Paolo P. Lava, Jun 13 2016

nonn

All terms appear to be multiples of 6.
Subset of A011775, A069235, A175200, A215142.
a(32) > 10^12. If p*r is a term, where p is prime and r is not divisible by p, then p^3*r is also a term. - Giovanni Resta, Jun 15 2016

			Divisors of 6 are 1, 2, 3 and 6 which sum to 12. The only exponential divisor is 6. Finally 12 / 6 = 2.
Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 which sum to 60. Exponential divisors are 6, 24 and their sum is 30. Finally 60 / 30 = 2.

Cf. A000203, A011775, A051377, A069235, A175200, A215142.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n,ok;
for n from 2 to q do a:=ifactors(n)[2]; b:=sort([op(divisors(n))]); c:=0;
for k from 2 to nops(b) do d:=ifactors(b[k])[2]; if nops(d)=nops(a) then
ok:=1; for j from 1 to nops(d) do if not type(a[j][2]/d[j][2],integer) then ok:=0; break; fi; od;
if ok=1 then c:=c+b[k]; fi;  fi; od; if sigma(n)=2*c  then print(n); fi; od; end: P(10^9);

Select[Range[10^6], 2 Times @@ Map[Sum[First[#]^d, {d, Divisors@ Last@ #}] &, FactorInteger@ #] == DivisorSigma[1, #] &] (* Michael De Vlieger, Jun 16 2016 *)

a(16)-a(31) from Giovanni Resta, Jun 15 2016

cp's OEIS Frontend

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

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A015706 Odd numbers k that divide phi(k)*sigma(k).

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A228104 Numbers of form 2^(2i-1)*3^j, with i,j > 0.

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A055196 Primitive numbers k that divide sigma(k)*phi(k).

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A066993 Integer of the form phi(n)*sigma(n)/n.

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A066994 Numbers k such that phi(k) divides k*sigma(k).

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A066995 Numbers k such that sigma(k) divides k*phi(k).

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A067575 Numbers k that divide phi(k)*bigomega(k).

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