A054024 -id:A054024 - cp's OEIS frontend

A196226 m such that A054024(m) (sum of divisors of m reduced modulo m) is 3 + m/2.

8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

John W. Layman, Sep 29 2011

nonn

This sequence appears to be identical to A073582 with its first term omitted and to A161344 with its first two terms omitted.
Conjectures.  (1) If m>=14 is a term of this sequence, then sigma(2,m) is congruent to 5 + m/2 modulo m;  (2) If m>=22 is a term of this sequence, then sigma(3,m) is congruent to 9 + m/2 modulo m;  If m>=38 is a term of this sequence, then sigma(4,m) is congruent to 17 + m/2 modulo m. (sigma(k,m) denotes the sum of the k-th powers of the divisors of m.)
Similar conjectures can be made about sigma(k,m) congruent to 2^k+1 + m/2 modulo m, for m a sufficiently large term of this sequence..
The even semiprimes (A100484) m= 2*p with p>3, with sigma(2*p)= 3+p (mod 2p), are a subsequence. - R. J. Mathar, Oct 02 2011
The terms in this sequence which are not even semiprimes are 8, 690, 12978, 176946, ... - R. J. Mathar, Aug 24 2023

Cf. A054024, A073582, A161344.

isA196226 := proc(n)
    sigmar := modp(numtheory[sigma](n),n) ;
    if sigmar = 3+n/2 then
        true;
    else
        false;
    end if;
end proc:
A196226 := proc(n)
     option remember;
     if n =1 then
        8;
    else
        for a from procname(n-1)+1 do
            if isA196226(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A196226(n),n=1..100) ; # R. J. Mathar, Aug 24 2023

lista(nn) = {for(n=1, nn, if ((sigma(n) % n) == (3 + n/2), print1(n, ", ")););} \\ Michel Marcus, Jul 12 2014

A337157 a(n) is the smallest m such that A054024(m) = prime(n), where A054024(m) is A000203(m) mod m, or -1 if there is no such m.

Original entry on oeis.org

20, 4, -1, 8, 21, 27, 39, 36, 57, 115, 32, 155, 63, 50, 129, 235, 265, 371, 305, 201, 98, 365, 237, 171, 245, 291, 485, 309, 325, 327, 128, 189, 279, 917, 1507, 1529, 242, 785, 489, 835, 865, 1211, 385, 605, 579, 324, 338, 2321, 669, 1115, 687, 1165, 399, 1936

Offset: 1

Michel Marcus, Jan 28 2021

sign

From Bernard Schott, Jan 28 2021: (Start)
a(n) > 0 when n <> 3.
Proof: When n = 1, 2, 4 then a(n) = 20, 4, 8; then, following Goldbach conjecture when p = prime(n) >=  11 with p-1 = p1 + p2, then sigma(p1*p2) = 1+p1+p2+p1*p2 == 1+p1+p2 = p (mod p1*p2); hence, for each n>= 5, a(n) exists and a(n) <= p1*p2 (see examples).
Now, when n = 3, no k is known such that sigma(k) == 5 (mod k)? (End)

			For prime(5) = 11, 11-1=3+7 with 3*7 = 21, so a(5) <= 21 and a(5) = 21.
For prime(6) = 13, 13-1=5+7 with 5*7 = 35, so a(6) <= 35 but sigma(27) = 40 == 13 (mod 27), hence a(6)=27.

Michel Marcus, Table of n, a(n) for n = 1..4000

Cf. A000203, A054024, A308150.

f(n) = sigma(n) % n; \\ A054024
a(n) = if (n==3, return (-1)); my(k=1, p=prime(n)); while (f(k) != p, k++); k;

A240092 Sequence of numbers starting at 1 and giving a new maximum record for sigma(n) modulo n (A054024), where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

1, 2, 4, 8, 10, 14, 16, 26, 32, 44, 50, 52, 60, 64, 76, 92, 105, 110, 128, 136, 152, 170, 184, 225, 230, 232, 248, 256, 296, 315, 336, 376, 410, 424, 470, 472, 484, 512, 568, 584, 592, 630, 656, 688, 752, 792, 848, 884, 944, 976, 988, 1012, 1024, 1072, 1136

Offset: 1

Michel Marcus, Apr 01 2014

nonn

If m is a power of 2, then sigma(m) = 2*m - 1 = m - 1, so sigma(m) == m-1 modulo m, thus giving a new record for A054024, hence A000079 is a subsequence.

			From the first terms of A054024 : 0, 1, 1, 3, 1, 0, 1, 7, 4, 8, 1, 4, 1, 10, ... we can see the records 0, 1, 3, 7, 8, 10, ... obtained for 1, 2, 4, 8, 10, ....

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000079, A000203, A054024.

DeleteDuplicates[Table[{n,Mod[DivisorSigma[1,n],n]},{n,1200}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jun 09 2024 *)

lista(nn) = {rec = -1; for (n=1, nn, sm = sigma(n) % n; if (sm > rec, rec = sm; print1(n, ", ");););}

A343242 Nonprime numbers k such that A054008(k) = A054024(k).

Original entry on oeis.org

1, 20, 196, 368, 650, 672, 780, 836, 1888, 2352, 3192, 11096, 17816, 20496, 30240, 51060, 84660, 130304, 979992, 1848964, 2291936, 3767100, 4526272, 8353792, 15126992, 15287976, 23569920, 33468416, 45532800, 74899952, 381236216, 623799776, 712023296, 1845991216

Offset: 1

Amiram Eldar, Apr 08 2021

nonn

If p is an odd prime then A054008(p) = A054024(p) = 2. Therefore, this sequence is restricted to nonprime numbers.

			20 is a term since it is a nonprime number and A054008(20) = A054024(20) = 2.

Cf. A000005, A000203, A054008, A054024.

Subsequences: A047728, A245782.

Select[Range[10^6], !PrimeQ[#] && Mod[#, DivisorSigma[0, #]] == Mod[DivisorSigma[1, #], #] &]

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

Original entry on oeis.org

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

A045770 Numbers k such that sigma(k) == 8 (mod k).

Original entry on oeis.org

1, 7, 10, 49, 56, 368, 836, 11096, 17816, 45356, 77744, 91388, 128768, 254012, 388076, 2087936, 2291936, 13174976, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216, 775397948, 3381872252, 4856970752, 6800228816, 8589344768, 44257207676, 114141404156, 1461083549696, 1471763808896, 2199013818368

Offset: 1

Dan Hoey

nonn

Every number of the form 2^(j-1)*(2^j - 9), where 2^j - 9 is prime, is a term. - Jon E. Schoenfield, Jun 02 2019

If m is a term of A045768 with gcd(m,3) = 1 and sigma(m) = 3*q*m + 2 for some integer q, then 3*m is a term of this sequence since sigma(3*m) = 4*q*(3*m) + 8. Some other large terms: 36893488108764397568, 877615520070055755776, 1700388548189538291286016, 85954979333046510417991676, 2081228720695521934665574252544. - Max Alekseyev, May 25 2025

Max Alekseyev, Table of n, a(n) for n = 1..41 (first 36 terms from Jud McCranie)

Cf. A054024, A045768, A045769, A088834, A076496.

q:= k-> nops(map(x-> x mod k, {8, numtheory[sigma](k)}))=1:
select(q, [$1..100000])[];  # Alois P. Heinz, Apr 07 2025

Select[Range[1000000], Mod[DivisorSigma[1, #] - 8, #] == 0 &] (* Pontus von Brömssen, Apr 07 2025 *)

isok(k) = Mod(sigma(k),k) == 8; \\ Pontus von Brömssen, Apr 07 2025

a(18)-a(26) from T. D. Noe, Apr 06 2011
Initial term 1 added and a(27)-a(31) from Donovan Johnson, Mar 01 2012
a(32)-a(34) from Giovanni Resta, Apr 02 2014
Term a(2)=7 inserted by Pontus von Brömssen, Apr 07 2025

A045768 Numbers k such that sigma(k) == 2 (mod k).

Original entry on oeis.org

1, 20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Dan Hoey

nonn

Equivalently, Chowla function of k is congruent to 1 (mod k).
If p=2^i-3 is prime, then 2^(i-1)*p is a term of the sequence. 650 is in the sequence, but is not of this form.
Terms k from a(2) to a(14) satisfy sigma(k) = 2*k + 2, implying that sigma(k) == 0 (mod k+1). It is not known if this holds in general, for there might be solutions of sigma(k)=3k+2 or 4k+2 or ... (Comments from Jud McCranie and Dean Hickerson, updated by Jon E. Schoenfield, Sep 25 2021 and by Max Alekseyev, May 23 2025).
k | sigma(k) produces the multiperfect numbers (A007691). It is an open question whether k | sigma(k) - 1 iff k is a prime or 1. It is not known if there exist solutions to sigma(k) = 2k+1.
Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - Benoit Cloitre, Feb 05 2002
Sequence seems to give nonprime k such that the numerator of the sum of the reciprocals of the divisors of k equals k+1 (nonprime k such that A017665(k)=k+1). - Benoit Cloitre, Apr 04 2002
For k > 1, composite numbers k such that A108775(k) = floor(sigma(k)/k) = sigma(k) mod k = A054024(k). Complement of primes (A000040) with respect to A230606. There are no numbers k > 2 such that sigma(x) = k*(x+1) has a solution. - Jaroslav Krizek, Dec 05 2013

			sigma(650) = 1302 == 2 (mod 650).

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

Amitabha Tripathi, A note on products of primes that differ by a fixed integer, Fibonacci Quart. 48 (2010), no. 2, 144-149.

Numbers k such that A054013(k)=1.

Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.

Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* Vincenzo Librandi, Mar 11 2014 *)

is(n)=sigma(n)%n==2 || n==1 \\ Charles R Greathouse IV, Mar 09 2014

More terms from Jud McCranie, Dec 22 1999.
a(11) from Donovan Johnson, Mar 01 2012
a(12) from Giovanni Resta, Apr 02 2014
a(13) from Jud McCranie, Jun 02 2019
Edited and a(14) from Jon E. Schoenfield confirmed by Max Alekseyev, May 23 2025

A159907 Numbers m with half-integral abundancy index, sigma(m)/m = k+1/2 with integer k.

Original entry on oeis.org

2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, 8583644160, 10200236032, 21857648640, 57575890944, 57629644800, 206166804480, 17116004505600, 1416963251404800, 15338300494970880, 75462255348480000, 88898072401645056, 301183421949935616, 6219051710415667200

Offset: 1

M. F. Hasler, Apr 25 2009

nonn

Obviously, all terms must be even (cf. formula), but e.g. a(9) and a(12) are not divisible by 3. See A007691 for numbers with integral abundancy.
Odd numbers and higher powers of 2 cannot be in the sequence; 6 is in A000396 and thus in A007691, and n=10,12,14,18,20,22 don't have integral 2*sigma(n)/n.
Conjecture: with number 1, multiply-anti-perfect numbers m: m divides antisigma(m) = A024816(m). Sequence of fractions antisigma(m) / m: {0, 0, 10, 2157, 2337, 13101, 4455356, ...}. - Jaroslav Krizek, Jul 21 2011
The above conjecture is equivalent to the conjecture that there are no odd multiply perfect numbers (A007691) greater than 1. Proof: (sigma(n)+antisigma(n))/n = (n+1)/2 for all n. If n is even then sigma(n)/n is a half-integer if and only if antisigma(n)/n is an integer. Since all members of this sequence are known to be even, the only way the conjecture can fail is if antisigma(n)/n is an integer, in which case sigma(n)/n is an integer as well. - Nathaniel Johnston, Jul 23 2011
These numbers are called hemiperfect numbers. See Numericana & Wikipedia links. - Michel Marcus, Nov 19 2017

			a(1) = 2 since sigma(2)/2 = (1+2)/2 = 3/2 is of the form k+1/2 with integer k=1.
a(2) = 24 is in the sequence since sigma(24)/24 = (1+2+3+4+6+8+12+24)/24 = (24+12+24)/24 = k+1/2 with integer k=2.

Max Alekseyev, Table of n, a(n) for n = 1..130
G. P. Michon, Multiperfect Numbers & Hemiperfect Numbers, Numericana.
Walter Nissen, Abundancy : Some Resources .
Project Euler, Problem 241: Perfection Quotients.
Wikipedia, Hemiperfect number

Cf. A000203, A088912, A141643 (k=2), A055153 (k=3), A141645 (k=4), A159271 (k=5).

isok(n) = denominator(sigma(n,-1)) == 2; \\ Michel Marcus, Sep 19 2015

forfactored(n=1,10^7, if(denominator(sigma(n,-1))==2, print1(n[1]", "))) \\ Charles R Greathouse IV, May 09 2017

from fractions import Fraction
from sympy import divisor_sigma as sigma
def aupto(limit):
  for k in range(1, limit):
    if Fraction(int(sigma(k, 1)), k).denominator == 2:
      print(k, end=", ")
aupto(3*10**4) # Michael S. Branicky, Feb 24 2021

A159907 = { n | 2*A000203(n) is in n*A005408 } = { n | A054024(n) = n/2 }

Terms a(20) onward from Max Alekseyev, Jun 05 2025

A229110 Sum of non-divisors of n reduced modulo n.

Original entry on oeis.org

0, 0, 2, 3, 4, 3, 6, 5, 5, 7, 10, 2, 12, 11, 6, 9, 16, 6, 18, 8, 10, 19, 22, 0, 19, 23, 14, 14, 28, 3, 30, 17, 18, 31, 22, 35, 36, 35, 22, 10, 40, 9, 42, 26, 12, 43, 46, 44, 41, 32, 30, 32, 52, 15, 38, 20, 34, 55, 58, 42, 60, 59, 22, 33, 46, 21, 66, 44, 42

Offset: 1

Jaroslav Krizek, Sep 22 2013

Numbers n such that a(n)=0 are: 1, 2, 24, 4320, 4680, ... (see A159907, conjecture by Jaroslav Krizek and further comments). - Michel Marcus, Sep 23 2013

Numbers n such that a(n)=n/2 are: 6, 28, 120, 496, 672, ... = A007691 \ {1}. - Michel Marcus, Sep 25 2013

Jaroslav Krizek, Table of n, a(n) for n = 1..10000.

Cf. A024816, A054024.

Haskell
```
a229110 n = mod (a024816 n) n
```

a(n) = lift(sum(i=1, n, if (n % i, Mod(i, n), 0))); \\ Michel Marcus, Sep 23 2013

a(n)=(n*(n+1)/2-sigma(n))%n \\ Charles R Greathouse IV, Sep 23 2013

a(n) = A024816(n) mod n.

A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 4, 4, 2, 4, 4, 3, 4, 3, 2, 5, 5, 4, 3, 4, 2, 4, 4, 5, 4, 5, 5, 4, 5, 5, 4, 4, 4, 5, 4, 4, 2, 5, 4, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 6, 5, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 5, 6, 5, 6, 6, 5, 4, 4, 5, 4, 4, 5, 6, 5, 5, 4, 6, 4, 4, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6

Offset: 1

Asher Auel, Jan 19 2000

The graph supports the conjecture that all numbers except 2 appear only a finite number of times. Sequences A000396, A005820, A027687, A046060 and A046061 give the n for which the abundancy sigma(n)/n is 2, 3, 4, 5 and 6, respectively. See A134639 for the number of n having abundancy greater than 2. - T. D. Noe, Nov 04 2007

T. D. Noe, Table of n, a(n) for n = 1..1600 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page
Eric Weisstein's World of Mathematics, Abundancy

Cf. A000203, A007691, A054024, A065997, A219545.

with(numtheory): for i while i < 33000 do
if sigma(i) mod i = 0 then print(sigma(i)/i) fi od;

for(n=1,1e7,if(denominator(k=sigma(n,-1))==1, print1(k", "))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A007691(n))/A007691(n)

More terms from Jud McCranie, Jul 09 2000

More terms from David Wasserman, Jun 28 2004

cp's OEIS Frontend

A196226 m such that A054024(m) (sum of divisors of m reduced modulo m) is 3 + m/2.

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A337157 a(n) is the smallest m such that A054024(m) = prime(n), where A054024(m) is A000203(m) mod m, or -1 if there is no such m.

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A240092 Sequence of numbers starting at 1 and giving a new maximum record for sigma(n) modulo n (A054024), where sigma(n) is the sum of divisors of n (A000203).

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Mathematica

PARI

A343242 Nonprime numbers k such that A054008(k) = A054024(k).

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Mathematica

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

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Haskell

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A045770 Numbers k such that sigma(k) == 8 (mod k).

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Maple

Mathematica

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A045768 Numbers k such that sigma(k) == 2 (mod k).

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Mathematica

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