A259307 -id:A259307 - 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

A259303 Numbers n such that there exists an x!=n that makes {x,x,n} an amicable multiset.

Original entry on oeis.org

1560, 7380, 20448, 20664, 22752, 23352, 25248, 26964, 35328, 51360, 100200, 104412, 108792, 135540, 156312, 192096, 232740, 252252, 278460, 310752, 362160, 401688, 492660, 543456, 548592, 588744, 659736, 732060, 767520, 826848, 998712, 999648, 1129440, 1129464

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {x,x,n} amicable iff sigma(x)=sigma(n)=x+x+n.
For the x values, see A259302.
If the condition x!=n were dropped, the terms from A005820 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1742

Cf. A005820, A259307, A259302, A259304, A259305, A259306, A125490.

isOK(n) = x=(sigma(n)-n)/2; x>0 && x!=n && denominator(x)==1 && sigma(x)==x+x+n;

A259302 Numbers n such that there exists an x!=n that makes {n,n,x} an amicable multiset.

Original entry on oeis.org

1740, 7776, 19260, 20640, 21384, 21924, 22428, 25830, 31440, 55968, 93024, 101010, 106140, 143910, 151164, 198792, 246510, 309582, 326196, 411138, 421596, 428256, 499464, 590112, 639288, 697158, 870552, 941094, 958716, 1060956, 1068210, 1087776, 1100640

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,x} amicable iff sigma(n)=sigma(x)=n+n+x.
For the x values, see A259303.
If the condition x!=n were dropped, the terms from A005820 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1689

Cf. A005820, A259307, A259303, A259304, A259305, A259306, A125490.

isOK(n) = x=sigma(n)-2*n; x>0 && x!=n && sigma(x)==n+n+x;

A259304 Numbers n such that there exists an x!=n that makes {n,n,n,x} an amicable multiset.

Original entry on oeis.org

796320, 1205820, 1277640, 1963080, 5154240, 8599500, 8694000, 9022860, 10820880, 12149280, 15845760, 17173800, 18023040, 19323360, 21360240, 22720320, 25494480, 27052200, 27335880, 28365480, 28574280, 28704060, 29211000, 30329640, 30373200, 30450420, 31433220

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,n,x} amicable iff sigma(n)=sigma(x)=n+n+n+x.
For the x values, see A259305.
If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259305, A259302, A259303, A259306, A036471.

isOK(n) = x=sigma(n)-3*n; x>0&x!=n&sigma(x)==3*n+x

A259305 Numbers n such that there exists an x!=n that makes {x,x,x,n} an amicable multiset.

Original entry on oeis.org

756000, 1220940, 1437480, 1973160, 5019840, 9058140, 9703260, 11058480, 11641680, 13085280, 17089920, 18117000, 19370880, 21282912, 21628080, 25295760, 25585560, 26936640, 27763560, 28191240, 29104200, 29203020, 30450420, 32839560, 33279120, 33398820

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {x,x,x,n} amicable iff sigma(x)=sigma(n)=x+x+x+n.
For the x values, see A259304.
If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259304, A259302, A259303, A259306, A036471.

isOK(n) = x=(sigma(n)-n)/3; x>0&x!=n&denominator(x)==1&sigma(x)==3*x+n

A259306 Numbers n such that there exists an x!=n that makes {n,n,x,x} an amicable multiset.

Original entry on oeis.org

1441440, 1860768, 3205440, 3378240, 3423420, 3914820, 4029480, 4437720, 5738040, 6093360, 6807240, 7136640, 7239120, 7551360, 9402120, 10204740, 12270720, 12405120, 12942720, 13495680, 14627340, 14725620, 15233400, 17166240, 18097800, 19158048, 23023440

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,x,x} amicable iff sigma(n)=sigma(x)=n+n+x+x.

If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259302, A259303, A259304, A259305, A036471.

isOK(n) = x=(sigma(n)-2*n)/2; x>0 && x!=n && denominator(x)==1 && sigma(x)==n+n+x+x

A255215 Numbers that belong to at least one amicable tuple.

Original entry on oeis.org

1, 220, 284, 1184, 1210, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 21168, 22200, 23940, 27312, 31284, 32136, 37380, 38940, 39480, 40068, 40608, 41412, 41952, 42168, 43890, 46368, 47124

Offset: 1

Jeppe Stig Nielsen, Feb 17 2015

nonn

Call a finite set {x_1, x_2, ..., x_k} of natural numbers (the x_i are pairwise distinct) an amicable k-tuple iff sigma(x_1)=sigma(x_2)=...=sigma(x_k)=x_1+x_2+...+x_k. Here sigma=A000203. For k=1, the only possible amicable one-tuple is {1}. For k=2 we get the classical amicable pairs (A063990). k=3 is amicable triples (A125490), k=4 amicable quadruples (A036471), and so on. A natural number n belongs to this sequence if and only if n is a member of some amicable k-tuple.
By definition, this sequence contains no duplicates.
For k<>2, an amicable k-tuple is not an aliquot cycle.

			1 belongs to this sequence because {1} is considered an amicable one-tuple.
284 belongs to this sequence because {220, 284} is an amicable pair.
2016 belongs to this sequence because {1980, 2016, 2556} is an amicable triple.
38940 is included in this sequence only once even if both {38940, 40068, 41952} and {38940, 40608, 41412} are amicable.
1000 does not belong to this sequence. To prove that, note that sigma(1000)=2340. Then find all x such that sigma(x)=2340, these are 792, 1000, 1062, 1305, 1611, 1945, 2339. Run through all subsets of 792, 1000, 1062, 1305, 1611, 1945, 2339 that include 1000 to verify that no such subset has a sum of 2340.
A tuple (or multiset) like {1560, 1740, 1740} where some element(s) are repeated, is not allowed here, and neither 1560 nor 1740 belongs to this sequence.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..361

Cf. A000203, A063990, A125490, A036471.

Cf. A259307 (duplicates allowed in tuple).

(notSubsetSum(desiredSum, searchSet) = { /* strongly inspired by is_A006037 function from A006037 */ local(t); /* return nonzero iff desiredSum is not the sum of a subset of searchSet */ setsearch( Set(searchSet), desiredSum ) & return /* equal to one element of searchSet */; while( #searchSet & searchSet[ #searchSet]>desiredSum, searchSet=vecextract(searchSet, "^-1")); desiredSum >= (t = sum(i=1, #searchSet, searchSet[i])) & return( desiredSum-t /* nonzero if desiredSum>t */ ); desiredSum > searchSet[ #searchSet] & ! notSubsetSum( desiredSum - searchSet[ #searchSet], searchSet=vecextract( searchSet, "^-1" )) & return; notSubsetSum( desiredSum, searchSet ) }); (othersWithSameSigma(n) = { s=sigma(n); [ x | x<-[1..s-1] , sigma(x)==s&&x!=n ] }); (is_A255215(x) = !notSubsetSum(sigma(x)-x, othersWithSameSigma(x)))

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

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A259303 Numbers n such that there exists an x!=n that makes {x,x,n} an amicable multiset.

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A259302 Numbers n such that there exists an x!=n that makes {n,n,x} an amicable multiset.

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A259304 Numbers n such that there exists an x!=n that makes {n,n,n,x} an amicable multiset.

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A259305 Numbers n such that there exists an x!=n that makes {x,x,x,n} an amicable multiset.

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A259306 Numbers n such that there exists an x!=n that makes {n,n,x,x} an amicable multiset.

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A255215 Numbers that belong to at least one amicable tuple.

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