A000203 -id:A000203 - cp's OEIS frontend

A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved.

0, 1, 2, 3, 4, 4, 6, 7, 8, 6, 10, 12, 12, 8, 14, 15, 16, 13, 18, 18, 20, 12, 22, 28, 24, 14, 26, 24, 28, 24, 30, 31, 32, 18, 34, 39, 36, 20, 38, 42, 40, 32, 42, 36, 44, 24, 46, 60, 48, 31, 50, 42, 52, 40, 54, 56, 56, 30, 58, 72, 60, 32, 62, 63, 64, 48

Offset: 0

Omar E. Pol, Jul 15 2014

nonn

Consider an irregular stepped pyramid with n steps. The base of the pyramid is equal to the symmetric representation of A024916(n), the sum of all divisors of all positive integers <= n. Two of the faces of the pyramid are the same as the representation of the n-th triangular numbers as a staircase. The total area of the pyramid is equal to 2*A024916(n) + A046092(n). The volume is equal to A175254(n). By definition a(2n-1) is A000203(n), the sum of divisors of n. Starting from the top a(2n-1) is also the total area of the horizontal part of the n-th step of the pyramid. By definition, a(2n) = A005843(n) = 2n. Starting from the top, a(2n) is also the total area of the irregular vertical part of the n-th step of the pyramid.
On the other hand the sequence also has a symmetric representation in two dimensions, see Example.
From Omar E. Pol, Dec 31 2016: (Start)
We can find the pyramid after the following sequences: A196020 --> A236104 --> A235791 --> A237591 --> A237593.
The structure of this infinite pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see the links).
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1, hence the sum of the areas of the terraces at the m-th level equals A000203(m).
Note that the stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the pyramid see A237593 and all its related sequences. (End)

			Illustration of initial terms:
----------------------------------------------------------------------
a(n)                             Diagram
----------------------------------------------------------------------
0    _
1   |_|\ _
2    \ _| |\ _
3     |_ _| | |\ _
4      \ _ _|_| | |\ _
4       |_ _|  _| | | |\ _
6        \ _ _|  _| | | | |\ _
7         |_ _ _|  _|_| | | | |\ _
8          \ _ _ _|  _ _| | | | | |\ _
6           |_ _ _| |    _| | | | | | |\ _
10           \ _ _ _|  _|  _|_| | | | | | |\ _
12            |_ _ _ _|  _|  _ _| | | | | | | |\ _
12             \ _ _ _ _|  _|  _ _| | | | | | | | |\ _
8               |_ _ _ _| |  _|  _ _|_| | | | | | | | |\ _
14               \ _ _ _ _| |  _| |  _ _| | | | | | | | | |\ _
15                |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |\ _
16                 \ _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |\
13                  |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | | |
18                   \ _ _ _ _ _| |  _|  _|    _ _| | | | | | | | | |
18                    |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | | |
20                     \ _ _ _ _ _ _| |      _| |  _ _ _| | | | | | |
12                      |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | | |
22                       \ _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | | |
28                        |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| | |
24                         \ _ _ _ _ _ _ _| |  _| |    _| |  _ _ _| |
14                          |_ _ _ _ _ _ _| | |  _|  _|  _| |  _ _ _|
26                           \ _ _ _ _ _ _ _| | |_ _|  _|  _| |
24                            |_ _ _ _ _ _ _ _| |  _ _|  _|  _|
28                             \ _ _ _ _ _ _ _ _| |  _ _|  _|
24                              |_ _ _ _ _ _ _ _| | |  _ _|
30                               \ _ _ _ _ _ _ _ _| | |
31                                |_ _ _ _ _ _ _ _ _| |
32                                 \ _ _ _ _ _ _ _ _ _|
...
a(n) is the total area of the n-th set of symmetric regions in the diagram.
.
From _Omar E. Pol_, Aug 21 2015: (Start)
The above structure contains a hidden pattern, simpler, as shown below:
Level                              _ _
1                                _| | |_
2                              _|  _|_  |_
3                            _|   | | |   |_
4                          _|    _| | |_    |_
5                        _|     |  _|_  |     |_
6                      _|      _| | | | |_      |_
7                    _|       |   | | |   |       |_
8                  _|        _|  _| | |_  |_        |_
9                _|         |   |  _|_  |   |         |_
10             _|          _|   | | | | |   |_          |_
11           _|           |    _| | | | |_    |           |_
12         _|            _|   |   | | |   |   |_            |_
13       _|             |     |  _| | |_  |     |             |_
14     _|              _|    _| |  _|_  | |_    |_              |_
15   _|               |     |   | | | | |   |     |               |_
16  |                 |     |   | | | | |   |     |                 |
...
The symmetric pattern emerges from the front view of the stepped pyramid.
Note that starting from this diagram A000203 is obtained as follows:
In the pyramid the area of the k-th vertical region in the n-th level on the front view is equal to A237593(n,k), and the sum of all areas of the vertical regions in the n-th level on the front view is equal to 2n.
The area of the k-th horizontal region in the n-th level is equal to A237270(n,k), and the sum of all areas of the horizontal regions in the n-th level is equal to sigma(n) = A000203(n). (End)
From _Omar E. Pol_, Dec 31 2016: (Start)
Illustration of the top view of the pyramid with 16 levels:
.
n   A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1      1   =      1      |_| | | | | | | | | | | | | | | |
2      3   =      3      |_ _|_| | | | | | | | | | | | | |
3      4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
4      7   =      7      |_ _ _|    _|_| | | | | | | | | |
5      6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
6     12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
7      8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
8     15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
9     13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
16    31   =     31      |_ _ _ _ _ _ _ _ _|
... (End)

Robert Price, Table of n, a(n) for n = 0..20000
Omar E. Pol, A pyramid related to the divisor function and other integers sequences
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A000203, A004125, A024916, A005843, A175254, A196020, A224880, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244970, A244971, A245093, A261350, A262626, A277437, A279387, A280223, A280295.

Table[If[EvenQ@ n, n, DivisorSigma[1, (n + 1)/2]], {n, 0, 65}] (* or *)
Transpose@ {Range[0, #, 2], DivisorSigma[1, #] & /@ Range[#/2 + 1]} &@ 65 // Flatten (* Michael De Vlieger, Dec 31 2016 *)
With[{nn=70},Riffle[Range[0,nn,2],DivisorSigma[1,Range[nn/2]]]] (* Harvey P. Dale, Aug 05 2024 *)

a(2*n-1) + a(2n) = A224880(n).

A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

1, 4, 8, 15, 21, 33, 41, 56, 69, 87, 99, 127, 141, 165, 189, 220, 238, 277, 297, 339, 371, 407, 431, 491, 522, 564, 604, 660, 690, 762, 794, 857, 905, 959, 1007, 1098, 1136, 1196, 1252, 1342, 1384, 1480, 1524, 1608, 1686, 1758, 1806, 1930, 1987, 2080, 2152

Offset: 1

Clark Kimberling

Row sums of triangle A128489. E.g., a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangle A128489. - Gary W. Adamson, Jun 03 2007
Row sums of triangle A134867. - Gary W. Adamson, Nov 14 2007
a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; see A072692. - M. F. Hasler, Nov 22 2007
Equals row sums of triangle A158905. - Gary W. Adamson, Mar 29 2009
n is prime if and only if a(n) - a(n-1) - 1 = n. - Omar E. Pol, Dec 31 2012
Also the alternating row sums of A236104. - Omar E. Pol, Jul 21 2014
a(n) is also the total number of parts in all partitions of the positive integers <= n into equal parts. - Omar E. Pol, Apr 30 2017
a(n) is also the total area of the terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Nov 04 2017
a(n) is also the area under the Dyck path described in the n-th row of A237593 (see example). - Omar E. Pol, Sep 17 2018
From Omar E. Pol, Feb 17 2020: (Start)
Convolution of A340793 and A000027.
Convolved with A340793 gives A000385. (End)
a(n) is also the number of cubic cells (or cubes) in the n-th level starting from the top of the stepped pyramid described in A245092. - Omar E. Pol, Jan 12 2022

			From _Omar E. Pol_, Aug 20 2021: (Start)
For n = 6 the sum of all divisors of the first six positive integers is [1] + [1 + 2] + [1 + 3] + [1 + 2 + 4] + [1 + 5] + [1 + 2 + 3 + 6] = 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33.
On the other hand the area under the Dyck path of the 6th diagram as shown below is equal to 33, so a(6) = 33.
Illustration of initial terms:                        _ _ _ _
                                        _ _ _        |       |_
                            _ _ _      |     |       |         |_
                  _ _      |     |_    |     |_ _    |           |
          _ _    |   |_    |       |   |         |   |           |
    _    |   |   |     |   |       |   |         |   |           |
   |_|   |_ _|   |_ _ _|   |_ _ _ _|   |_ _ _ _ _|   |_ _ _ _ _ _|
.
    1      4        8          15           21             33         (End)

Hardy and Wright, "An introduction to the theory of numbers", Oxford University Press, fifth edition, p. 266.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Vaclav Kotesovec, Plot of (a(n) - Pi^2*n^2/12) / (n*log(n)^(2/3)) for n = 2..100000.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916.
Peter Polm, C# program for A024916.
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.

Partial sums of A000203.
Cf. A056550, A104471(2*n-1, n), A123229, A128489, A000217, A134867, A072692, A158905, A237593, A245092, A006218, A222548, A092406, A160664.
Cf. A000385, A010766, A340793.

a024916 n = sum $ map (\k -> k * div n k) [1..n]
-- Reinhard Zumkeller, Apr 20 2015

[(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Mar 15 2019

A024916 := proc(n)
    add(numtheory[sigma](k),k=0..n) ;
end proc: # Zerinvary Lajos, Jan 11 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      numtheory[sigma](n)+a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 12 2019

Table[Plus @@ Flatten[Divisors[Range[n]]], {n, 50}] (* Alonso del Arte, Mar 06 2006 *)
Table[Sum[n - Mod[n, m], {m, n}], {n, 50}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2006 *)
a[n_] := Sum[DivisorSigma[1, k], {k, n}]; Table[a[n], {n, 51}] (* Jean-François Alcover, Dec 16 2011 *)
Accumulate[DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Mar 13 2014 *)

A024916(n)=sum(k=1,n,n\k*k) \\ M. F. Hasler, Nov 22 2007

A024916(z) = { my(s,u,d,n,a,p); s = z*z; u = sqrtint(z); p = 2; for(d=1, u, n = z\d - z\(d+1); if(n<=1, p=d; break(), a = z%d; s -= (2*a+(n-1)*d)*n/2); ); u = z\p; for(d=2, u, s -= z%d); return(s); } \\ See the link for a nicely formatted version. - P. L. Patodia (pannalal(AT)usa.net), Jan 11 2008

A024916(n)={my(s=0,d=1,q=n);while(dPeter Polm, Aug 18 2014

A024916(n)={ my(s=n^2, r=sqrtint(n), nd=n, D); for(d=1, r, (1>=D=nd-nd=n\(d+1)) && (r=d-1) && break; s -= n%d*D+(D-1)*D\2*d); s - sum(d=2, n\(r+1), n%d)} \\ Slightly optimized version of Patodia's code. - M. F. Hasler, Apr 18 2015
(C#) See Polm link.

def A024916(n): return sum(k*(n//k) for k in range(1,n+1)) # Chai Wah Wu, Dec 17 2021

from math import isqrt
def A024916(n): return (-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 21 2023

[sum(sigma(k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Mar 15 2019

From Benoit Cloitre, Apr 28 2002: (Start)
a(n) = n^2 - A004125(n).
Asymptotically a(n) = n^2*Pi^2/12 + O(n*log(n)). (End)
G.f.: (1/(1-x))*Sum_{k>=1} x^k/(1-x^k)^2. - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{m=1..n} (n - (n mod m)). - Roger L. Bagula and Gary W. Adamson, Oct 06 2006
a(n) = n^2*Pi^2/12 + O(n*log(n)^(2/3)) [Walfisz]. - Charles R Greathouse IV, Jun 19 2012
a(n) = A000217(n) + A153485(n). - Omar E. Pol, Jan 28 2014
a(n) = A000292(n) - A076664(n), n > 0. - Omar E. Pol, Feb 11 2014
a(n) = A078471(n) + A271342(n). - Omar E. Pol, Apr 08 2016
a(n) = (1/2)*(A222548(n) + A006218(n)). - Ridouane Oudra, Aug 03 2019
From Greg Dresden, Feb 23 2020: (Start)
a(n) = A092406(n) + 8, n>3.
a(n) = A160664(n) - 1, n>0. (End)
a(2*n) = A326123(n) + A326124(n). - Vaclav Kotesovec, Aug 18 2021
a(n) = Sum_{k=1..n} k * A010766(n,k). - Georg Fischer, Mar 04 2022

A003601 Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 105

Offset: 1

N. J. A. Sloane, Mira Bernstein

Sometimes called arithmetic numbers.
Generalized (sigma_r)-numbers are numbers j for which sigma_r(j)/sigma_0(j) = c^r. Sigma_r(j) denotes the sum of the r-th powers of the divisors of j; c,r are positive integers. The numbers in this sequence are sigma_1-numbers; those in A140480 are sigma_2-numbers. - Ctibor O. Zizka, Jul 14 2008
{a(n)} = union A175678 and A175679 where A175678 = numbers m such that the arithmetic mean Ad(m) of divisors of m and the arithmetic mean Ah(m) of numbers h < m such that gcd(h,m) = 1 are both integers and A175679 = numbers m such that the arithmetic mean Ad(m) of the divisors of m and the arithmetic mean Ak(m) of the numbers k <= m are both integers. - Jaroslav Krizek, Aug 07 2010
All odd primes (A065091) are arithmetic numbers. - Wesley Ivan Hurt, Oct 04 2013
A069928(n) = number of arithmetic numbers not greater than n. - Reinhard Zumkeller, Jul 28 2014
A102187(n) divides a(n) for a(n) = 1, 6, 140, 270, 672, ... A007340. - Thomas Ordowski, Oct 24 2014
The quotients sigma(j)/tau(j) are in A102187. - Bernard Schott, Jun 07 2017

			Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - _Bernard Schott_, Jun 07 2017

R. K. Guy, Unsolved Problems in Number Theory, B2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Paul T. Bateman, Paul Erdős, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer (1981). In Knopp, M.I. ed., Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics. 899. Springer-Verlag., pp. 197-220.
Antonio M. Oller-Marcén, On arithmetic numbers, arXiv:1206.1823 [math.NT], 2012.
O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
Wikipedia, Arithmetic number.

Complement is A049642.
Cf. A000005, A000203, A054025, A001599, A007340, A140480, A102187.
Cf. A245644, A245656, A069928. Nonprimes are in A023883.

a:=Filtered([1..110],n->Sigma(n) mod Tau(n)=0);; Print(a); # Muniru A Asiru, Jan 25 2019

a003601 n = a003601_list !! (n-1)
a003601_list = filter ((== 1) . a245656) [1..]
-- Reinhard Zumkeller, Jul 28 2014, Dec 31 2013, Jan 06 2012

with(numtheory); t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod tau(n) = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # corrected by Wesley Ivan Hurt, Oct 03 2013

Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] (* Stefan Steinerberger, Apr 03 2006 *)

is(n)=sigma(n)%numdiv(n)==0 \\ Charles R Greathouse IV, Jul 10 2012

from sympy import divisors, divisor_count
[n for n in range(1,10**5) if not sum(divisors(n)) % divisor_count(n)] # Chai Wah Wu, Aug 05 2014

a(n) ~ n. - Charles R Greathouse IV, Jul 10 2012

A245656(a(n)) = 1. - Reinhard Zumkeller, Jul 28 2014

David W. Wilson, Oct 15 1996, points out that 30 was missing.

More terms from Stefan Steinerberger, Apr 03 2006

A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

Original entry on oeis.org

1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828

Offset: 1

David W. Wilson

nonn

The quotient A020492(n)/A002088(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 or zeta(2)^2 [~2.705808084277845]. - Labos Elemer, Sep 20 2004, corrected by Charles R Greathouse IV, Jun 20 2012
If 2^p-1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because when p = 2 we get m = 3 and phi(3) divides sigma(3) and for p > 2, phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sigma(m)/phi(m) = 4 is an integer. So for each n, A133028(n) = 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht, Nov 28 2005
Phi and sigma are both multiplicative functions and for this reason if m and n are coprime and included in this sequence then m*n is also in this sequence. - Enrique Pérez Herrero, Sep 05 2010
The quotients sigma(n)/phi(n) are in A023897. - Bernard Schott, Jun 06 2017
There are 544768 balanced numbers < 10^14. - Jud McCranie, Sep 10 2017
a(975807) = 419998185095132. - Jud McCranie, Nov 28 2017

			sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.

D. Chiang, "N's for which phi(N) divides sigma(N)", Mathematical Buds, Chap. VI pp. 53-70 Vol. 3 Ed. H. D. Ruderman, Mu Alpha Theta 1984.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jud McCranie, 670314 balanced numbers (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)

Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).

Positions of 0's in A063514.

[ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // Klaus Brockhaus, Nov 09 2008

Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]
(* Second program: *)
Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* Michael De Vlieger, Nov 28 2017 *)

select(n->sigma(n)%eulerphi(n)==0,vector(10^4,i,i)) \\ Charles R Greathouse IV, Jun 20 2012

from sympy import totient, divisor_sigma
print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # Indranil Ghosh, Jul 06 2017

from math import prod
from itertools import count, islice
from sympy import factorint
def A020492_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        f = factorint(m)
        if not prod(p**(e+2)-p for p,e in f.items())%(m*prod((p-1)**2 for p in f)):
            yield m
A020492_list = list(islice(A020492_gen(),20)) # Chai Wah Wu, Aug 12 2024

More terms from Farideh Firoozbakht, Nov 28 2005

A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600

Offset: 1

Matthew Conroy

Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996
With respect to the comment above, neither sequence is subsequence of the other. - Ivan N. Ianakiev, Feb 11 2020
Also n such that sigma_{-1}(n) > sigma_{-1}(m) for all m < n, where sigma_{-1}(n) is the sum of the reciprocals of the divisors of n. - Matthew Vandermast, Jun 09 2004
Ramanujan (1997, Section 59; written in 1915) called these numbers "generalized highly composite." Alaoglu and Erdős (1944) changed the terminology to "superabundant." - Jonathan Sondow, Jul 11 2011
Alaoglu and Erdős show that: (1) n is superabundant => n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2 >= e_3 >= ... >= e_p (and e_p is 1 unless n=4 or n=36); (2) if q < r are primes, then | e_r - floor(e_q*log(q)/log(r)) | <= 1; (3) q^{e_q} < 2^{e_2+2} for primes q, 2 < q <= p. - Keith Briggs, Apr 26 2005
It follows from Alaoglu and Erdős finding 1 (above) that, for n > 7, a(n) is a Zumkeller Number (A083207); for details, see Proposition 9 and Corollary 5 at Rao/Peng link (below). - Ivan N. Ianakiev, Feb 11 2020
See A166735 for superabundant numbers that are not highly composite, and A189228 for superabundant numbers that are not colossally abundant.
Pillai called these numbers "highly abundant numbers of the 1st order". - Amiram Eldar, Jun 30 2019

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad. Publ., 2002 (see pp. 19-21).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

D. Kilminster, Table of n, a(n) for n = 1..2000 (extends to n = 8436 in the comments; first 500 terms from T. D. Noe)
A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273-275.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
Christian Axler, Inequalities involving the primorial counting function, arXiv:2406.04018 [math.NT], 2024. See p. 12.
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Tibor Burdette and Ian Stewart, Counterexamples to a Conjecture by Alaoglu and Erdős, arXiv:2009.03306 [math.NT], 2020.
Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, INTEGERS 11 (2011), #A33.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
François Clément and Stefan Steinerberger, On the largest singular vector of the Redheffer matrix, arXiv:2502.09489 [math.NT], 2025. See p. 2.
P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants (Text in French), Bulletin de la S. M. F., tome 103 (1975), pp. 65-90.
F. Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
A. Morkotun, On the increase of Gronwall function value at the multiplication of its argument by a prime, arXiv preprint arXiv:1307.0083 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv preprint arXiv:1211.2147 [math.NT], 2012.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Walter Nissen, Abundancy : Some Resources.
T. D. Noe, First 500 superabundant numbers.
T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped).
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Superabundant number.

Almost the same as A077006.
The colossally abundant numbers A004490 are a subsequence, as are A023199.
Subsequence of A025487; apart from a(3) = 4 and a(7) = 36, a subsequence of A102750.
Cf. A000203, A002093, A002182.
Cf. A112974 (number of superabundant numbers between colossally abundant numbers).
Cf. A091901 (Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality), A192884 (non-superabundant and the reverse of Robin's inequality).

a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
(* Second program: convert all 8436 terms in b-file into a list of terms: *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, ?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]}; Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ] (* _Michael De Vlieger, May 08 2018 *)

print1(r=1);forstep(n=2,1e6,2,t=sigma(n,-1);if(t>r,r=t;print1(", "n))) \\ Charles R Greathouse IV, Jul 19 2011

a(n+1) <= 2*a(n). - A.H.M. Smeets, Jul 10 2021

Name edited by Peter Munn, Mar 13 2019

A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

Original entry on oeis.org

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864

Offset: 1

N. J. A. Sloane

Let sigma_m(n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives (2,2)-perfect numbers.
Even values of these are 2^(p-1) where 2^p-1 is a Mersenne prime (A000043 and A000668). No odd superperfect numbers are known. Hunsucker and Pomerance checked that there are no odd ones below 7 * 10^24. - Jud McCranie, Jun 01 2000
The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - Omar E. Pol, Feb 29 2008
The sum of divisors of a(n) is the n-th Mersenne prime A000668(n), provided that there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Largest proper divisor of A072868(n) if there are no odd superperfect numbers. - Omar E. Pol, Apr 25 2008
This sequence is a divisibility sequence if there are no odd superperfect numbers. - Charles R Greathouse IV, Mar 14 2012
For n>1, sigma(sigma(a(n))) + phi(phi(a(n))) = (9/4)*a(n). - Farideh Firoozbakht, Mar 02 2015
The term "super perfect number" was coined by Suryanarayana (1969). He and Kanold (1969) gave the general form of even superperfect numbers. - Amiram Eldar, Mar 08 2021

			sigma(sigma(4))=2*4, so 4 is in the sequence.

Dieter Bode, Über eine Verallgemeinerung der vollkommenen Zahlen, Dissertation, Braunschweig, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B9, pp. 99-100.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, pp. 110-111.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, pp. 38-42.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

P. Bundschuh, Aufgabe 601, Elem. Math., Vol. 24 (1969), p. 69; alternative link.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
G. G. Dandapat, J. L. Hunsucker, and Carl Pomerance, Some new results on odd perfect numbers, Pacific J. Math. Volume 57, Number 2 (1975), 359-364.
A. Hoque and H. Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett. 3, No. 3, 249-253 (2014).
J. L. Hunsucker and Carl Pomerance, There are no odd superperfect number less than 7*10^24, Indian J. Math., Vol. 17 (1975), pp. 107-120.
H.-J. Kanold, Über "Super perfect numbers", Elem. Math., Vol. 24 (1969), pp. 61-62; alternative link.
Graham Lord, Even Perfect and Superperfect Numbers, Elem. Math., Vol. 30 (1975), pp. 87-88.
H. G. Niederreiter, Solution of Aufgabe 601, Elem. Math., Vol. 25 (1970), pp. 66-67; alternative link.
Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
D. Suryanarayana, Super perfect numbers, Elem. Math., Vol. 24 (1969), pp. 16-17; alternative link.
D. Suryanarayana, There is no superperfect number of the form p^(2*alpha), Elem. Math., Vol. 28 (1973), pp. 148-150; alternative link.
László Tóth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siófok, Hungary.
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Eric Weisstein's World of Mathematics, Superperfect Number.
Wikipedia, Superperfect number.
Tomohiro Yamada, On finiteness of odd superperfect numbers, Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 259-274.

Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A072868.

sigma = DivisorSigma[1, #]&;
For[n = 2, True, n++, If[sigma[sigma[n]] == 2 n, Print[n]]] (* Jean-François Alcover, Sep 11 2018 *)

is(n)=sigma(sigma(n))==2*n \\ Charles R Greathouse IV, Nov 20 2012

from itertools import count, islice
def A019279_gen(): # generator of terms
    return (n for n in count(1) if divisor_sigma(divisor_sigma(n)) == 2*n)
A019279_list = list(islice(A019279_gen(),6)) # Chai Wah Wu, Feb 18 2022

a(n) = (1 + A000668(n))/2, if there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - Omar E. Pol, Apr 25 2008
a(n) = 2^A090748(n), if there are no odd superperfect numbers. - Ivan N. Ianakiev, Sep 04 2013

a(8)-a(9) from Jud McCranie, Jun 01 2000

Corrected by Michel Marcus, Oct 28 2017

A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

Original entry on oeis.org

0, 1, 3, 4, 7, 6, 15, 8, 12, 13, 31, 12, 63, 18, 18, 24, 127, 14, 255, 20, 39, 48, 511, 24, 28, 84, 24, 48, 1023, 32, 2047, 32, 54, 176, 42, 40, 4095, 258, 144, 56, 8191, 38, 16383, 68, 36, 800, 32767, 48, 60, 31, 252, 132, 65535, 30, 91, 72, 528, 1302, 131071, 44, 262143, 2736, 60, 104, 126, 96, 524287, 304, 774, 42, 1048575, 72, 2097151, 4356, 42

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A323244, A323247, A323248, A324118, A324543 (Möbius transform), A324396, A324823.

Cf. A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Array[If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 75] (* Michael De Vlieger, Apr 21 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323243(n) = if(1==n, 0, sigma(A156552(n)));

\\ For computing terms a(n), with n > ~4000 use Hans Havermann's factorization file https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt"); \\ First read it in as a PARI-vector.
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1))); \\ Then play sigma
\\ Antti Karttunen, Mar 15 2019

from sympy import divisor_sigma, primepi, factorint
def A323243(n): return divisor_sigma(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).
a(n) = 2*A156552(n) - A323244(n).
a(n) = A323247(n) - A323248(n).
From Antti Karttunen, Mar 12 2019: (Start)
a(A000040(n)) = A000225(n).
a(n) = Sum_{d|n} A324543(d).
For n > 1, a(2*A246277(n)) = A324118(n).
gcd(a(n), A156552(n)) = A324396(n).
A000035(a(n)) = A324823(n).
(End)

A014567 Numbers k such that k and sigma(k) are relatively prime, where sigma(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 36, 37, 39, 41, 43, 47, 49, 50, 53, 55, 57, 59, 61, 63, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 97, 98, 100, 101, 103, 107, 109, 111, 113, 115, 119, 121, 125, 127, 128, 129, 131, 133

Offset: 1

Eric W. Weisstein

Related to "solitary numbers": n is solitary if there is no other integer m such that sigma(m)/m = sigma(n)/n.
It is easy to show that if n and sigma(n) are relatively prime then n is solitary. But the converse is not true; for example, 18, 45, 48 and 52 are solitary. Probably also 10, 14, 15, 20, 22 and many others are solitary, but I do not think that will ever be proved. - Dean Hickerson
From Daniel Forgues, Jun 23 2009: (Start)
Union of unit, primes and Duffinian numbers.
Duffinian numbers (A003624) are the composite numbers (including, among others, the proper prime powers) for which (n, sigma(n)) = 1. (End)
A009194(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2013
These numbers satisfy (denominator of sigma(n)/n) = n. - Michel Marcus, Oct 27 2013
The asymptotic density of this sequence is 0 (Dressler, 1974; Luca, 2007). - Amiram Eldar, Jul 23 2020
If m*n is in this sequence and gcd(m,n) = 1, then m and n are both in this sequence. - Jianing Song, Aug 07 2022

			sigma(21) = 1 + 3 + 7 + 21 = 32 is relatively prime to 21, so 21 is in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
C. W. Anderson and D. Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84, 65-66, 1977.
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.
Andrew Feist, Fun with the sigma(n) function, Missouri Journal of Mathematical Sciences 15:3 (2003), pp. 173-177.
P. A. Loomis, New families of solitary numbers, J. Algebra and Applications, 14 (No. 9, 2015), #1540004 (6 pages).
Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
Eric Weisstein's World of Mathematics, Solitary Number.

Cf. A003624.
Cf. A069059 (complement).
Includes A000961 as a subsequence.

a014567 n = a014567_list !! (n-1)
a014567_list = filter ((== 1) . a009194) [1..]
-- Reinhard Zumkeller, Mar 23 2013

lst={};Do[d=DivisorSigma[1, n];If[GCD[d, n]==1, AppendTo[lst, n]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
Select[Range[150],CoprimeQ[#,DivisorSigma[1,#]]&] (* Harvey P. Dale, Jan 23 2015 *)

is(n)=gcd(n,sigma(n))==1 \\ Charles R Greathouse IV, Feb 13 2013

from math import gcd
from sympy import divisor_sigma
def ok(n): d = divisor_sigma(n, 1); return gcd(n, d) == 1
print([k for k in range(1, 134) if ok(k)]) # Michael S. Branicky, Mar 28 2022

a(n) << n log n. Can this be improved? - Charles R Greathouse IV, Feb 13 2013

a(n) >> n log log log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

More terms from Labos Elemer

A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k), 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 2, 3, 4, 3, 6, 4, 7, 5, 9, 8, 7, 6, 7, 15, 12, 14, 6, 12, 11, 21, 20, 21, 12, 12, 8, 15, 33, 28, 35, 18, 24, 8, 15, 22, 45, 44, 49, 30, 36, 16, 15, 13, 30, 66, 60, 77, 42, 60, 24, 30, 13, 18, 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12, 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28

Offset: 1

Omar E. Pol, Jan 20 2013

Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014
From Omar E. Pol, Jul 10 2021: (Start)
The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.
The base of the tower is the symmetric representation of A024916(n).
The height of the tower is equal to A000041(n-1).
The surface area of the tower is equal to A345023(n).
The volume (or the number of cubes) of the tower equals A066186(n).
The volume represents the n-th term of the convolution of A000203 and A000041, that is A066186(n).
Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
The tower is an object of the family of the stepped pyramid described in A245092.
T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower.
T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206.
The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
Therefore the set of all partitions of n >= 1 has an associated tower.
The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.
Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)

			Triangle begins:
------------------------------------------------------
    n| k    1   2   3   4   5   6   7   8   9  10
------------------------------------------------------
    1|      1;
    2|      1,  3;
    3|      2,  3,  4;
    4|      3,  6,  4,  7;
    5|      5,  9,  8,  7,  6;
    6|      7, 15, 12, 14,  6, 12;
    7|     11, 21, 20, 21, 12, 12,  8;
    8|     15, 33, 28, 35, 18, 24,  8, 15;
    9|     22, 45, 44, 49, 30, 36, 16, 15, 13;
   10|     30, 66, 60, 77, 42, 60, 24, 30, 13, 18;
...
The sum of row 10 is [30 + 66 + 60 + 77 + 42 + 60 + 24 + 30 + 13 + 18] = A066186(10) = 420.
.
For n = 10 the calculation of the row 10 is as follows:
    k    A000203         T(10,k)
    1       1   *  30   =   30
    2       3   *  22   =   66
    3       4   *  15   =   60
    4       7   *  11   =   77
    5       6   *   7   =   42
    6      12   *   5   =   60
    7       8   *   3   =   24
    8      15   *   2   =   30
    9      13   *   1   =   13
   10      18   *   1   =   18
                 A000041
.
From _Omar E. Pol_, Jul 13 2021: (Start)
For n = 10 we can see below three views of two associated polycubes called here "prism of partitions" and "tower". Both objects contain the same number of cubes (that property is valid for n >= 1).
        _ _ _ _ _ _ _ _ _ _
  42   |_ _ _ _ _          |
       |_ _ _ _ _|_        |
       |_ _ _ _ _ _|_      |
       |_ _ _ _      |     |
       |_ _ _ _|_ _ _|_    |
       |_ _ _ _        |   |
       |_ _ _ _|_      |   |
       |_ _ _ _ _|_    |   |
       |_ _ _      |   |   |
       |_ _ _|_    |   |   |
       |_ _    |   |   |   |
       |_ _|_ _|_ _|_ _|_  |                             _
  30   |_ _ _ _ _        | |                            | | 30
       |_ _ _ _ _|_      | |                            | |
       |_ _ _      |     | |                            | |
       |_ _ _|_ _ _|_    | |                            | |
       |_ _ _ _      |   | |                            | |
       |_ _ _ _|_    |   | |                            | |
       |_ _ _    |   |   | |                            | |
       |_ _ _|_ _|_ _|_  | |                           _|_|
  22   |_ _ _ _        | | |                          |   |  22
       |_ _ _ _|_      | | |                          |   |
       |_ _ _ _ _|_    | | |                          |   |
       |_ _ _      |   | | |                          |   |
       |_ _ _|_    |   | | |                          |   |
       |_ _    |   |   | | |                          |   |
       |_ _|_ _|_ _|_  | | |                         _|_ _|
  15   |_ _ _ _      | | | |                        | |   |  15
       |_ _ _ _|_    | | | |                        | |   |
       |_ _ _    |   | | | |                        | |   |
       |_ _ _|_ _|_  | | | |                       _|_|_ _|
  11   |_ _ _      | | | | |                      | |     |  11
       |_ _ _|_    | | | | |                      | |     |
       |_ _    |   | | | | |                      | |     |
       |_ _|_ _|_  | | | | |                     _| |_ _ _|
   7   |_ _ _    | | | | | |                    |   |     |   7
       |_ _ _|_  | | | | | |                   _|_ _|_ _ _|
   5   |_ _    | | | | | | |                  | | |       |   5
       |_ _|_  | | | | | | |                 _| | |_ _ _ _|
   3   |_ _  | | | | | | | |               _|_ _|_|_ _ _ _|   3
   2   |_  | | | | | | | | |           _ _|_ _|_|_ _ _ _ _|   2
   1   |_|_|_|_|_|_|_|_|_|_|          |_ _|_|_|_ _ _ _ _ _|   1
.
             Figure 1.                       Figure 2.
         Front view of the                 Lateral view
        prism of partitions.               of the tower.
.
.                                      _ _ _ _ _ _ _ _ _ _
                                      |   | | | | | | | |_|   1
                                      |   | | | | | |_|_ _|   2
                                      |   | | | |_|_  |_ _|   3
                                      |   | |_|_    |_ _ _|   4
                                      |   |_ _  |_  |_ _ _|   5
                                      |_ _    |_  |_ _ _ _|   6
                                          |_    | |_ _ _ _|   7
                                            |_  |_ _ _ _ _|   8
                                              |           |   9
                                              |_ _ _ _ _ _|  10
.
                                             Figure 3.
                                             Top view
                                           of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 10 in colexicographic order (cf. A026792, A211992). The area of the diagram is 10*42 = A066186(10) = 420. Note that the diagram can be interpreted also as the front view of a right prism whose volume is 1*10*42 = 420 equaling the volume and the number of cubes of the tower that appears in the figures 2 and 3.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions. In this case the mentioned area equals A000070(10-1) = 97.
The connection between these two associated objects is a representation of the correspondence divisor/part described in A338156. See also A336812.
The sum of the volumes of both objects equals A220909.
For the connection with the table of A338156 see also A340035. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened)
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
Omar E. Pol, Illustration of the prism, the tower and the 10th row of the triangle

Row sums give A066186.
Column 1 is A000041.
Leading diagonals 1-5: A000203, A000203, A074400, A272027, A274535.
Companion of A221530.
Cf. A000070, A000203, A026792, A027293, A135010, A138137, A176206, A182703, A220909, A211992, A221649, A236104, A237270, A237271, A237593, A245092, A245093, A245095, A245099, A262626, A336811, A336812, A338156, A339278, A340035, A340583, A340584, A345023, A346741.

nrows=12; Table[Table[DivisorSigma[1,k]PartitionsP[n-k],{k,n}],{n,nrows}] // Flatten (* Paolo Xausa, Jun 17 2022 *)

T(n,k)=sigma(k)*numbpart(n-k) \\ Charles R Greathouse IV, Feb 19 2013

T(n,k) = sigma(k)*p(n-k) = A000203(k)*A027293(n,k).

T(n,k) = A245093(n,k)*A027293(n,k).

A053866 Parity of A000203(n), the sum of the divisors of n; a(n) = 1 when n is a square or twice a square, 0 otherwise.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Mar 29 2000

Also parity of A001227, the number of odd divisors of n. - Omar E. Pol, Apr 04 2016
Also parity of A000593, the sum of odd divisors of n. - Omar E. Pol, Apr 05 2016
Characteristic function of A028982. - Antti Karttunen, Sep 25 2017
It appears that this is also the parity of A067742, the number of middle divisors of n. - Omar E. Pol, Mar 18 2018
Also parity of the deficiency of n (A033879) and of the abundance of n (A033880). - Omar E. Pol, Nov 02 2024

Antti Karttunen, Table of n, a(n) for n = 1..65536
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq. 16 (2013) #13.1.8.
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for characteristic functions

Essentially same as A093709.

Cf. A000203, A000593, A001227, A028982, A033879, A033880, A286357, A292583, A295896, A019590, A025441, A145393, A359548 (Dirichlet inverse).

A053866:= (n -> numtheory[sigma](n) mod 2):
seq (A053866(n), n=0..104); # Jani Melik, Jan 28 2011

Mod[DivisorSigma[1,Range[110]],2] (* Harvey P. Dale, Sep 04 2017 *)

{a(n) = if( n<1, 0, issquare(n) || issquare(2*n))} /* Michael Somos, Apr 12 2004 */

from sympy.ntheory.primetest import is_square
def A053866(n): return int(is_square(n) or is_square(n<<1)) # Chai Wah Wu, Jan 09 2023

a(n) = A000203(n) mod 2. a(n)=1 iff n>0 is a square or twice a square.
Multiplicative with a(2^e)=1, a(p^e)=1 if e even, 0 otherwise.
a(n) = A093709(n) if n>0.
Dirichlet g.f.: zeta(2s)(1+2^-s). - Michael Somos, Apr 12 2004
a(n) = A001157(n) mod 2. - R. J. Mathar, Apr 02 2011
a(n) = floor(sqrt(n)) + floor(sqrt(n/2)) - floor(sqrt(n-1))-floor(sqrt((n-1)/2)). - Enrique Pérez Herrero, Oct 15 2013
a(n) = A000035(A000203(n)). - Omar E. Pol, Oct 26 2013
a(n) = A063524(A286357(n)) = A063524(A292583(n)). - Antti Karttunen, Sep 25 2017
a(n) = A295896(A156552(n)). - Antti Karttunen, Dec 02 2017
a(n) = Sum_{ m: m^2|n } A019590(n/m^2). - Andrey Zabolotskiy, May 07 2018
G.f.: (theta_3(x) + theta_3(x^2))/2 - 1. - Ilya Gutkovskiy, May 23 2019
Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020

More terms from James Sellers, Apr 08 2000

Alternative description added to the name by Antti Karttunen, Sep 25 2017

cp's OEIS Frontend

A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved.

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A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

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A003601 Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).

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A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

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A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

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