A007691 -id:A007691 - cp's OEIS frontend

A046764 Sum of the 4th powers of the divisors of n is divisible by n.

1, 34, 84, 156, 364, 492, 1092, 3444, 5617, 6396, 11234, 22468, 33628, 44772, 67404, 100884, 157276, 190978, 292084, 435708, 437164, 471828, 549687, 569772, 709937, 742612, 763912, 876252, 986076, 1099374, 1118480, 1289484, 1311492, 1419874

Offset: 1

Labos Elemer

nonn

Compare with multiply perfect numbers, A007691. Here Sum[ divisors ] is replaced by Sum[ 4th powers of divisors ].

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

			n=84, Sigma[ 4,84 ] = Sum(d^4) = 53771172 = 640133*84 = 640133*n;
n=5617, Sigma[ 4,5617 ] = 995446331475844 = 5617*17722083332, a multiple of n.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A001159, A007691.

Do[If[Mod[DivisorSigma[4, n], n]==0, Print[n]], {n, 1, 2*10^6}]
Select[Range[1500000],Divisible[DivisorSigma[4,#],#]&] (* Harvey P. Dale, Jun 25 2014 *)

is(n)=sigma(n, 4)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

Mod[ Sigma [ 4, n ], n ]=0.

More terms from Robert G. Wilson v, Jun 09 2000

A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

Original entry on oeis.org

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920, 156036748944739017459105792, 3638193973609385308194865152

Offset: 1

Labos Elemer

nonn

Colton proves that perfect numbers (A000396) cannot be refactorable.

			k = 45532800 is a term since s0 = d(k) = 384, s1 = sigma(k) = 571963392, and the four quotients s1/s0 = 474300, (k * s0)/s1 = 96, s1/k = 4 and k/s0 = 118580 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..305
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

Cf. A000005, A000203, A000396, A046985, A001599, A007691.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n * d, s] && Divisible[s, d] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !((k * d) % s) && !(s % d) && !(k % d);} \\ Amiram Eldar, May 09 2024

Let s1 = sigma(k) = A000203(k) be the sum of divisors of k and s0 = d(k) = A000005(k) be the number of divisors of k. Then, k is a term if s1/s0, (k * s0)/s1, s1/k, and k/s0 are all integers.

a(7)-a(13) from Donovan Johnson, Apr 09 2010

Edited and a(14)-a(16) added by Amiram Eldar, May 09 2024

A056006 Numbers k such that k | sigma(k) + 2.

Original entry on oeis.org

1, 3, 10, 136, 32896, 2147516416

Offset: 1

Robert G. Wilson v, Jul 24 2000

n | sigma(n) gives the multi-perfect numbers A007691, n | sigma(n)+1 if n is a power of 2 (A000079).
This contains A191363 as subsequence, so for any Fermat prime F(k) = 2^2^k+1, the triangular number A000217(2^2^k)=(F(k)-1)*F(k)/2 is in this sequence. See also A055708 which is identical up to the first term. - M. F. Hasler, Oct 02 2014
a(7) > 10^13. - Giovanni Resta, Jul 13 2015
a(7) > 10^18. - Max Alekseyev, May 27 2025

Cf. A000203, A045768, A055708.

Do[If[Mod[DivisorSigma[1, n]+2, n]==0, Print[n]], {n, 1, 7*10^8}]

for(n=1,5e9,if((sigma(n)+2)%n==0,print1(n", "))) \\ Charles R Greathouse IV, Jun 01 2011

a(6) from Charles R Greathouse IV, Jun 01 2011

Edited by M. F. Hasler, Oct 02 2014

A066284 a(n) = A066135(4*n).

Original entry on oeis.org

34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 386, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194

Offset: 1

Labos Elemer, Dec 11 2001

nonn

a(n) <= 2p, where p = A002586(4n) is the least prime factor of (1 + 16^n). (See the Comment in A066135.) - Jonathan Sondow, Nov 23 2012

			First 3 terms correspond to entries of other sequences as follows: a(1)=A046763(2), a(2)=A055712(2), a(3)=A055716(2).
From _Michael De Vlieger_, Jul 17 2017: (Start)
First position of values, with observations pertaining to values for 1 <= n <= 3000:
    Value   Position   Observations:
    --------------------------------
       34     1        All odd.
       84     2        In A047235.
      194     6        In A017593.
      228    12
      386    36
     1282    72
     1538   144
     3084   288
   147468   576
     1956   864
  1046532  1152
    24578  2304
     3252  2880
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A066135, A046761, A046762, A046763, A055709-A055717, A007691, A001159.

Table[m = 2; While[Mod[DivisorSigma[4 n, m], m] > 0, m++]; m, {n, 66}] (* Michael De Vlieger, Jul 17 2017 *)

a(n) = {n *= 4; my(m = 2); while (sigma(m, n) % m, m++); m;} \\ Michel Marcus, Oct 02 2016

a(n) = Min{x : sigma_4n(x) mod x = 0, x > 1}

A066289 Numbers k such that k divides DivisorSigma(2*j-1, k) for all j; i.e., all odd-power-sums of divisors of k are divisible by k.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 31998395520, 796928461056000, 212517062615531520, 680489641226538823680000, 13297004660164711617331200000, 1534736870451951230417633280000, 6070066569710805693016339910206758877366156437562171488352958895095808000000000

Offset: 1

Labos Elemer, Dec 12 2001

nonn

Tested for each k and j < 200. Otherwise the proof for all j seems laborious, since the number of divisors of terms of sequence rapidly increases: {1, 4, 16, 24, 96, 96, 2304, ...}.
Tested for each k and j <= 1000. - Thomas Baruchel, Oct 10 2003
The given terms have been tested for all j. - Don Reble, Nov 03 2003
This is a proper subset of the multiply perfect numbers A007691. E.g., 8128 from A007691 is not here because its remainder at Sigma[odd,8128]/8128 division is 0 or 896 depending on odd exponent.

Cf. A066135, A066284, A007691, A066290.

DivisorSigma(2*j-1, k)/k is an integer for all j = 1, 2, 3, ..., 200, ...

The following numbers belong to the sequence, but there may be missing terms in between: 796928461056000 (also belongs to A046060); 212517062615531520 (also belongs to A046060); 680489641226538823680000 (also belongs to A046061); 13297004660164711617331200000 (also belongs to A046061). - Thomas Baruchel, Oct 10 2003

Extended to 13 confirmed terms by Don Reble, Nov 04 2003. There is a question whether there are other members below a(13). However, there are none in Achim's list of multiperfect numbers (see A007691); Richard C. Schroeppel has suggested that that list is complete to 10^70 - if so, a(1..12) are correct; as for a(13), Rich says there's only "an epsilon chance that some undiscovered MPFN lies in the gap." So it is very likely to be correct. - Don Reble

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

A082901 a(n) = A082895(n)-A000203(n); the distance from sigma(n) to that multiple of n which is closest to sigma(n), positive terms for cases where the closest multiple is after sigma(n), and negative terms where it is before sigma(n). In case of ties, a positive term is selected.

Original entry on oeis.org

0, 1, -1, 1, -1, 0, -1, 1, -4, 2, -1, -4, -1, 4, 6, 1, -1, -3, -1, -2, 10, 8, -1, 12, -6, 10, -13, 0, -1, -12, -1, 1, -15, 14, -13, 17, -1, 16, -17, -10, -1, -12, -1, 4, 12, 20, -1, 20, -8, 7, -21, 6, -1, -12, -17, -8, -23, 26, -1, 12, -1, 28, 22, 1, -19, -12, -1, 10, -27, -4, -1, 21, -1, 34, 26, 12, -19, -12, -1, -26, -40, 38, -1, 28

Offset: 1

Labos Elemer, Apr 22 2003

			n=2: sigma(2)=3, the closest even numbers to 3 are 2 and 4, we choose 4 to get a positive difference, thus a(2) = 4-3 = 1.
n=28: sigma(28) = 56, thus a multiple of 28 which is closest to 28 is 28, so the difference is zero. Positions of zeros for this sequence is given by the multiply perfect numbers, A007691.
When n is a prime p > 2, sigma(p) = p+1, thus the multiple of p closest to p+1 is p, so difference is -1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A082893-A082900, A000203, A007691.

Table[n*Floor[(Floor[n/2]+DivisorSigma[1, n])/n]- DivisorSigma[1, n], {n, 1, 100}]

a(n)=my(s=sigma(n));s\/n*n-s \\ Charles R Greathouse IV, Feb 15 2013

A082901(n) = { my(s=sigma(n),  a = ((s\n)*n)-s, b = ((1+(s\n))*n)-s); if(b <= abs(a), b, a); }; \\ Antti Karttunen, Oct 01 2018

a(n) = n*floor[(floor(n/2)+sigma(n))/n] - sigma(n).

Definition clarified and the example section edited by Antti Karttunen, Sep 25 2018

A173438 Number of divisors d of number n such that d does not divide sigma(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 4, 1, 4, 3, 2, 1, 2, 2, 2, 3, 0, 1, 4, 1, 5, 2, 2, 3, 8, 1, 2, 3, 4, 1, 4, 1, 3, 4, 2, 1, 7, 2, 5, 2, 4, 1, 4, 3, 4, 3, 2, 1, 6, 1, 2, 5, 6, 3, 4, 1, 4, 2, 6, 1, 10, 1, 2, 5, 3, 3, 4, 1, 8, 4, 2, 1, 6, 3, 2, 2, 5, 1, 6, 2, 3, 3, 2, 2, 6, 1, 5

Offset: 1

Jaroslav Krizek, Feb 18 2010

nonn

a(n) = 0 for multiply-perfect numbers (A007691).

			For n = 12, a(12) = 3; sigma(12) = 28, divisors of 12: 1, 2, 3, 4, 6, 12; d does not divide sigma(n) for 3 divisors d: 3, 6 and 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000203, A007691, A009194, A073802, A286570.

A173438 := proc(n)
    local sd,a;
    sd := numtheory[sigma](n) ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(sd,d) <> 0 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015

Table[DivisorSum[n, 1 &, ! Divisible[DivisorSigma[1, n], #] &], {n, 98}] (* Michael De Vlieger, Oct 08 2017 *)

A173438(n) = (numdiv(n) - numdiv(gcd(sigma(n), n))); \\ (See PARI-code in A073802) - Antti Karttunen, Oct 08 2017

a(n) = A000005(n) - A073802(n).

a(n) = tau(n) - tau(gcd(n,sigma(n))). - Antti Karttunen, Oct 08 2017

A189000 Bi-unitary multiperfect numbers.

Original entry on oeis.org

1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240, 342720, 523776, 1028160, 1528800, 6168960, 7856640, 7983360, 14443520, 22932000, 23569920, 43330560, 44553600, 51979200, 57657600, 68796000, 133660800, 172972800, 779688000, 1476304896, 2339064000, 6840038400

Offset: 1

R. J. Mathar, Apr 15 2011

nonn

All entries greater than 1 are even [Hagis].
14443520 is the first (only?) composite term not divisible by 3.  Excluding the factor p=3, all composite terms <= 172972800 have nonincreasing exponents in the factorization (sorted by primes). - D. S. McNeil, Apr 15 2011
Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - Tomohiro Yamada, Apr 15 2017
McNeil's observation about exponents does not hold in general. Indeed, a(41) = 2^8 * 3^5 * 5^2 * 7 * 11 * 13^2 * 17. - Giovanni Resta, Apr 15 2017
a(43) > 4.66*10^12. - Giovanni Resta, Sep 07 2018
We include 1 here, although this is not "multi"-perfect. - R. J. Mathar, Sep 08 2018

			n=120 divides A188999(120)=360.
n=90 divides A188999(90)=180.
n=672 divides A188999(672)=2016.

Giovanni Resta, Table of n, a(n) for n = 1..42
Peter Hagis, Bi-Unitary amicable and multiperfect numbers, Fib. Quart. 25 (2) (1987) 144-151
Pentti Haukkanen and V. Sitaramaiah, Bi-unitary multiperfect numbers, I, Notes Number Theory Discrete Math. 26 (1) (2020) 93-171.
Michel Marcus, Unexhaustive list of terms
C. R. Wall, Bi-unitary perfect numbers, Proc. Amer. Math. Soc. 33 (1) (1972) 39-42.
Tomohiro Yamada, Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024.

Cf. A007691 (analog for sigma).

Cf. A188999 (bi-unitary sigma), A318175, A318781 (the k coefficients).

bsig[n_] := If[n == 1, 1, Block[{p, e}, Product[{p, e} = pe; (p^(e + 1) - 1)/(p - 1) - If[EvenQ[e], p^(e/2), 0], {pe, FactorInteger[n]}]]]; Select[Range[10^5], Mod[bsig[#], #] == 0 &] (* Giovanni Resta, Apr 15 2017 *)

a188999(n) = {my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = ! frac(a188999(n)/n); \\ Michel Marcus, Sep 03 2018

{n | A188999(n)}.

a(18)-a(27) by D. S. McNeil, Apr 15 2011
a(28)-a(31) from Giovanni Resta, Apr 15 2017
a(1)=1 inserted by Giovanni Resta, Sep 07 2018

A214842 Anti-multiply-perfect numbers. Numbers n for which sigma(n)/n is an integer, where sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

1, 2, 5, 8, 41, 56, 77, 946, 1568, 2768, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714, 455879783074, 528080296234, 673223621664, 4054397778846, 4437083907194, 4869434608274, 6904301600914, 7738291969456

Offset: 1

Paolo P. Lava, Mar 08 2013

nonn

A073930 and A073931 are subsets of this sequence.
Like A007691 but using sigma*(n) (A066417) instead of sigma(n) (A000203).
Tested up to 167786. Additional terms are 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714 but there may be missing terms among them.

			Anti-divisors of 77 are 2, 3, 5, 9, 14, 17, 22, 31, 51. Their sum is 154 and 154/77=2.

Cf. A000203, A007691, A066272, A066417, A073930, A073931.

A214842:= proc(q) local a,k,n;
for n from 1 to q do
  a:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
  if type(a/n,integer) then print(n); fi; od; end:
A214842(10^10);

a066417[n_Integer] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; a214842[n_Integer] := Select[Range[n], IntegerQ[a066417[#]/#] &];
a214842[1200] (* Michael De Vlieger, Aug 08 2014 *)

sad(n) = vecsum(select(t->n%t && tA066417
isok(n) = denominator(sad(n)/n) == 1; \\ Michel Marcus, Oct 12 2019

A214842 = [n for n in range(1,10**4) if not (sum([d for d in range(2,n,2) if n%d and not 2*n%d])+sum([d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]])) % n]
# Chai Wah Wu, Aug 12 2014

Verified there are no missing terms up to a(24) by Donovan Johnson, Apr 13 2013
a(25)-a(27) by Jud McCranie, Aug 31 2019
a(28)-a(32) by Jud McCranie, Oct 10 2019

cp's OEIS Frontend

A046764 Sum of the 4th powers of the divisors of n is divisible by n.

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A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

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A056006 Numbers k such that k | sigma(k) + 2.

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A066284 a(n) = A066135(4*n).

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A066289 Numbers k such that k divides DivisorSigma(2*j-1, k) for all j; i.e., all odd-power-sums of divisors of k are divisible by k.

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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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A082901 a(n) = A082895(n)-A000203(n); the distance from sigma(n) to that multiple of n which is closest to sigma(n), positive terms for cases where the closest multiple is after sigma(n), and negative terms where it is before sigma(n). In case of ties, a positive term is selected.

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A214842 Anti-multiply-perfect numbers. Numbers n for which sigma(n)/n is an integer, where sigma(n) is the sum of the anti-divisors of n.