A007691 -id:A007691 - cp's OEIS frontend

A046986 Multiply perfect numbers that are also harmonic numbers but are not arithmetic numbers.

28, 496, 8128, 2178540, 33550336, 142990848, 459818240, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128

Offset: 1

Labos Elemer

nonn

			k = 2178540 is a term since s0 = d(k) = 216 and s1 = sigma(k) = 8714160, s1/s0 = 8714160/216 = 121030/3 is not an integer, and (k * s0)/s1 = (2178540 * 216)/8714160 = 54 and s1/k = 8714160/2178540 = 4 are integers.

Amiram Eldar, Table of n, a(n) for n = 1..207

In A007691 and A001599 but not in A003601.

Cf. A000005, A000203, A046985, A046987.

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, s1 | (k * s0), but s1 is not divisible by s0.

a(12)-a(17) from Donovan Johnson, Nov 30 2008

Edited and a(18)-a(21) added by Amiram Eldar, May 09 2024

A088843 Number of divisors of multiply-perfect numbers.

Original entry on oeis.org

1, 4, 6, 16, 10, 24, 14, 96, 96, 80, 216, 320, 26, 384, 480, 288, 576, 224, 34, 1920, 2304, 1056, 480, 896, 38, 960, 1344, 3456, 1620, 5280, 6336, 3888, 10368, 9216, 2816, 3584, 6480, 16384, 5400, 9600, 62, 25344, 12960, 41472, 110592, 32256, 62208, 51840, 48384

Offset: 1

Labos Elemer, Nov 05 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1600 (calculated from the b-file at A007691)

Cf. A000005, A007691.

s = {}; Do[If[Divisible[DivisorSigma[1, n], n], AppendTo[s, DivisorSigma[0, n]]], {n, 1, 10^6}]; s (* Amiram Eldar, Jul 20 2019 *)

a(n) = A000005(A007691(n)).

More terms from Michel Marcus, Sep 19 2013

Data corrected by Amiram Eldar, Jul 20 2019

A225110 Numbers m such that S = Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of m in increasing order, and q the smallest integer 1 < q <= tau(m) for m > 1; and a(1) = 1.

Original entry on oeis.org

1, 6, 18, 28, 42, 54, 66, 78, 102, 114, 120, 126, 138, 162, 174, 180, 186, 196, 198, 222, 234, 246, 258, 282, 294, 306, 318, 342, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 496, 498, 522, 534, 546, 558, 582, 594, 606, 618, 642, 654, 666, 672, 678, 702, 714

Offset: 1

Michel Lagneau, Apr 28 2013

nonn

By convention, for n = 1, a(1) = 1 with q = 1.
The corresponding q are 1, 4, 4, 6, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 15, 4, 6, 4,...
Properties of this sequence:
q = tau(n) if n = 1, 6, 28, 120, 496,... is a multiply-perfect numbers: n divides sigma(n) (see A007691). This numbers are in the sequence.
S = 2 for a majority of n
S = 3 for n = 120, 180, 672, 1890, 8460, 9540,...
S = 4 for n = 30240, 32760, 90720,...

			18 is in the sequence because the divisors of 18 are 1, 2, 3, 6, 9 and 18 => 1 + 1/2 + 1/3 + 1/6 = 2.
28 is in the sequence because 28 is a multiply-perfect numbers: the divisors are 1, 2, 4, 7, 14, 28 and 1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2.
From _Michael De Vlieger_, Sep 15 2017: (Start)
Records k and first positions n of records of q that pertain to a(n) for values less than or equal to 10^7:
   i     k        n       a(n)
  ----------------------------
   1     1        1         1
   2     4        2         6
   3     6        4        28
   4    10       39       496
   5    14      608      8128
   6    15       16       180
   7    16       11       120
   8    17     1543     20482
   9    18     2521     33345
  10    20      629      8415
  11    21      145      1890
  12    22    30824    407715
  13    24       52       672
  14    26     2908     38430
  15    28     3034     40128
  16    30     1917     25410
  17    34    96461   1274100
  18    35     1544     20496
  19    43    61026    806190
  20    45     7839    103530
  21    54     5512     72800
  22    58    74184    979992
  23    69     6871     90720
  24    77   270202   3571050
  25    80    39625    523776
  26    96     2284     30240
  27   216   164870   2178540
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A007691.

with(numtheory): for n from 1 to 1000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 then ii:=1: printf(`%d, `,n):else fi:od:od:

Select[Range@ 714, Function[n, AnyTrue[If[n > 1, Rest@ #, #] &@ FoldList[Plus, 1/Divisors@ n], IntegerQ]]] (* Michael De Vlieger, Sep 15 2017 *)

isok(k) = if (k==1, return(1)); my(d=divisors(k), s=1); for (i=2, #d, s += 1/d[i]; if (denominator(s)==1, return(1));); \\ Michel Marcus, Feb 22 2025

Name edited by Michel Marcus, Jun 13 2025

A227302 Numbers m such that m divides sigma(2*m).

Original entry on oeis.org

1, 3, 12, 14, 60, 248, 336, 2160, 2340, 4064, 13104, 15120, 16380, 261888, 1089270, 4455360, 8714160, 10213632, 11784960, 16775168, 22766400, 45981824, 71495424, 98532480, 229909120, 689727360, 738152448, 4291822080, 4294934528, 5100118016, 7091219520

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

If m belongs to the sequence, then sigma(2*m)/m is an integer, so sigma(2*m)/(2*m) is either an integer or half of an integer, so 2*m is either perfect, multiperfect or hemiperfect. - Michel Marcus, Jul 09 2013

Jinyuan Wang, Table of n, a(n) for n = 1..54

Cf. A000203, A000396, A007691, A159907, A232702.

Cf. A141643, A055153, A141645, A159271, A160678. (hemiperfect numbers)

is(n)=sigma(2*n)%n==0 \\ Charles R Greathouse IV, Nov 04 2016

A227470 Least k such that n divides sigma(n*k).

Original entry on oeis.org

1, 3, 2, 3, 8, 1, 4, 7, 10, 4, 43, 2, 9, 2, 8, 21, 67, 5, 37, 6, 20, 43, 137, 5, 149, 9, 34, 1, 173, 4, 16, 21, 27, 64, 76, 22, 73, 37, 6, 3, 163, 10, 257, 43, 6, 137, 281, 11, 52, 76, 67, 45, 211, 17, 109, 4, 49, 173, 353, 2, 169, 8, 32, 93, 72, 27, 401, 67

Offset: 1

Alex Ratushnyak, Jul 12 2013

nonn

Theorem: a(n) always exists.

Proof: If n is a power of a prime, say n = p^a, then, by Euler's generalization of Fermat's little theorem and the multiplicative property of sigma, one can take k = x^(p^a-p^(a-1)-1) where x is a different prime from p. Similarly. if n = p^a*q^b, then take k = x^(p^a-p^(a-1)-1)*y^(q^b-q^(b-1)-1) where {x,y} are primes different from {p,q}. And so on. These k's have the desired property, and so there is always at least one candidate for the minimal k. - N. J. A. Sloane, May 01 2016

			Least k such that 9 divides sigma(9*k) is k = 10: sigma(90) = 234 = 9*26. So a(9) = 10.
Least k such that 89 divides sigma(89*k) is k = 1024: sigma(89*1024) = 184230 = 89*2070. So a(89) = 1024.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Indices of 1's: A007691.
Cf. A000203, A227302, A227303, A097018.
See A272349 for the sequence [n*a(n)]. - N. J. A. Sloane, May 01 2016

A227470 := proc(n)
    local k;
    for k from 1 do
        if modp(numtheory[sigma](k*n),n) =0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, May 06 2016

lknds[n_]:=Module[{k=1},While[!Divisible[DivisorSigma[1,k*n],n],k++];k]; Array[lknds,70] (* Harvey P. Dale, Jul 10 2014 *)

a227470(n) = {k=1; while(sigma(n*k)%n != 0, k++); k} \\ Michael B. Porter, Jul 15 2013

a(n) = A272349(n)/n. - R. J. Mathar, May 06 2016

A263928 Integers m such that sigma(m)^2 is divisible by m.

Original entry on oeis.org

1, 6, 24, 28, 120, 224, 234, 270, 496, 588, 600, 672, 864, 1080, 1521, 1638, 1782, 2016, 3724, 4320, 4680, 5733, 6048, 6200, 6552, 7128, 8128, 11172, 11466, 15872, 17280, 18144, 18600, 18620, 21600, 22932, 26208, 26460, 27000, 30240, 32640, 32760, 33516, 35640

Offset: 1

Paolo P. Lava, Oct 30 2015

nonn

Previous name was: "Numbers such that the product of the sum of their divisors and the sum of the reciprocals of their divisors is an integer".
The multiply-perfect numbers (A007691) are a subset of this sequence.
This is a subsequence of A175200. - Michel Marcus, Nov 03 2015
Alternative definition: Numbers m such that Sum_{i = 1..k} (sigma(m) - d_i) / d_i is an integer, where d_i are the k divisors of m. - Paolo P. Lava, Mar 23 2017

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Their sum is sigma(24) = 60 while the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/12 + 1/24 = 5/2. Finally 60 * 5/2 = 150.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A000203, A007691, A017665, A017666, A175200, A263983.

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if type(sigma(n)^2/n,integer) then print(n);
fi; od; end: P(10^6);

Select[Range[36000],Divisible[DivisorSigma[1,#]^2,#]&] (* Harvey P. Dale, Jul 05 2023 *)

isok(n) = (sigma(n)^2 % n) == 0; \\ Michel Marcus, Nov 03 2015

New name from Michel Marcus, Nov 03 2015

A318996 a(n) = Sum_{d|n} (sigma(n) mod d).

Original entry on oeis.org

0, 1, 1, 4, 1, 0, 1, 11, 5, 11, 1, 9, 1, 13, 13, 26, 1, 10, 1, 8, 17, 17, 1, 16, 7, 19, 18, 0, 1, 28, 1, 57, 19, 23, 22, 34, 1, 25, 23, 24, 1, 41, 1, 65, 45, 29, 1, 57, 9, 68, 25, 75, 1, 39, 25, 25, 29, 35, 1, 88, 1, 37, 74, 120, 29, 37, 1, 91, 31, 24, 1, 103

Offset: 1

Jaroslav Krizek, Sep 07 2018

nonn

			For n = 4; a(4) = (7 mod 1) + (7 mod 2) + (7 mod 4) = 0 + 1 + 3 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Carlos Rivera, Puzzle 1065. A larger integer than 45 such that ..., The Prime Puzzles and Problems Connection.

Cf. A000005, A000040, A000203, A007691, A008578, A300657.

[&+[SumOfDivisors(n) mod d: d in Divisors(n)] : n in [1..1000]]

a[n_] := Block[{s = DivisorSigma[1, n]}, DivisorSum[n, Mod[s, #] &]]; Array[a, 72] (* Giovanni Resta, Sep 07 2018 *)

a(n) = my(sn = sigma(n)); sumdiv(n, d, sn % d); \\ Michel Marcus, Sep 07 2018

from sympy import divisors
def a(n): divs = divisors(n); s = sum(divs); return sum(s%d for d in divs)
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Nov 27 2021

a(A007691(n)) = 0.
a(A000040(n)) = 1.
a(A008578(n)) = tau(n) - 1.
a(n) = n for numbers 4, 45, 6048, 14421, ...

A325639 Numbers n for which A091255(n, sigma(n)) = n.

Original entry on oeis.org

1, 6, 28, 120, 312, 428, 456, 496, 504, 672, 760, 6552, 8128, 30240, 31452, 32760, 429240, 523776, 2178540, 5009850, 7505976, 23569920, 33550336, 45532800, 142990848, 186076800, 379975680

Offset: 1

Antti Karttunen, May 21 2019

Numbers n for which A000203(n) = A048720(n, k) for some k. The value of k for the initial terms is: 1, 2, 2, 7, 3, 3, 6, 2, 5, 3, 3, 6, 2, 4, 6, 4, 6, 7, 4, 3, 6, 4, 2, 4, 4, 7, 7, ...
Conjecture: all terms after the initial one are even. If this is true, then there are no odd perfect numbers.
A007691(11) = 2178540 is the first term of A007691 which is not present in this sequence.

Index entries for sequences where any odd perfect numbers must occur

Fixed points of A325632 and A325634.
Cf. A000203, A007691, A048720, A091255.
Cf. A000396, A325638 (subsequences).

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));
isA325639(n) = (A325634(n)==n);

A341524 Number of prime factors in A017666(n), counted with multiplicity: a(n) = bigomega(n) - bigomega(gcd(n, sigma(n))).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 0, 1, 1, 1, 5, 1, 1, 2, 4, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 6, 2, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 1, 2, 1, 1, 4, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 0

Offset: 1

Antti Karttunen, Feb 17 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A001222, A009194, A017666, A341523.
Cf. A007691 (positions of zeros).
Cf. A341608 (applied onto prime shift array A246278).

Table[PrimeOmega[n] - PrimeOmega[GCD[n, DivisorSigma[1, n]]], {n, 1, 100}] (* Amiram Eldar, Feb 17 2021 *)

A341524(n) = (bigomega(n) - bigomega(gcd(n, sigma(n))));

a(n) = A001222(A017666(n)).

a(n) = A001222(n) - A341523(n).

A353764 Numbers k for which A353749(sigma(k)) is a multiple of A353749(k), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 18, 20, 24, 28, 30, 32, 40, 60, 72, 84, 90, 108, 120, 128, 200, 216, 224, 234, 252, 360, 384, 496, 600, 640, 672, 864, 936, 1080, 1120, 1152, 1170, 1488, 1800, 1920, 2016, 2176, 3200, 3360, 3456, 4320, 4464, 4680, 5148, 5600, 5760, 6048, 6528, 6552, 8128, 9600, 10080, 10880, 14976, 16800, 17280

Offset: 1

Antti Karttunen, May 10 2022

nonn

Question: Are there any odd terms after the initial one? See A353789, A353796, A353797.

Cf. A000010, A000203, A006872, A062401, A064989, A336549, A336550, A353757, A353761, A353766 (characteristic function), A353789, A353796, A353797.
Positions of 1's in A353762. Cf. also A353765.
Subsequence of A353759. Cf. A007691 (a subsequence).

f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[20000], Divisible[s[DivisorSigma[1, #]], s[#]] &] (* Amiram Eldar, May 10 2022 *)

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
isA353764(n) = { my(s=sigma(n)); !(A353749(s)%A353749(n)); };

cp's OEIS Frontend

A046986 Multiply perfect numbers that are also harmonic numbers but are not arithmetic numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A088843 Number of divisors of multiply-perfect numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A225110 Numbers m such that S = Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of m in increasing order, and q the smallest integer 1 < q <= tau(m) for m > 1; and a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A227302 Numbers m such that m divides sigma(2*m).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A227470 Least k such that n divides sigma(n*k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A263928 Integers m such that sigma(m)^2 is divisible by m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A318996 a(n) = Sum_{d|n} (sigma(n) mod d).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python