A065764 -id:A065764 - cp's OEIS frontend

A350072 a(n) = sigma(n^2) / gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

1, 7, 13, 31, 31, 91, 57, 127, 121, 31, 133, 403, 183, 133, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 427, 1093, 589, 871, 403, 993, 2047, 133, 2149, 1767, 3751, 1407, 889, 2379, 3937, 1723, 1729, 1893, 4123, 3751, 3871, 2257, 6643, 2801, 781, 3991, 1891, 2863, 7651, 589, 2413, 4953, 6097, 3541, 12493

Offset: 1

Antti Karttunen, Dec 12 2021

nonn

Conjecture: There are no 1's after the initial term. Remark: If there were some k = x^2 > 1, for which a(x) = 1, then sigma(k) would be a divisor of A003961(k). In other words, d = A350073(k) = A064989(sigma(k)) would be a divisor of k. Then, if that divisor were also a unitary divisor [with gcd(d,k/d) = 1], it would need to satisfy the equation sigma(k) = sigma(d) * sigma(k/d) = sigma(A064989(sigma(k))) * sigma(k/A064989(sigma(k))), because sigma is a multiplicative function. (Minor correction by Antti Karttunen, Jul 11 2023)

Note that if d = A064989(sigma(k)) were a unitary divisor of a square k, then sigma(k) would also be a square, the cases which are quite rare (see A008848 and A336547). Also compare to A349756. - Antti Karttunen, Jul 24 2022

Cf. A000203, A003961, A008848, A064989, A065764, A349162, A350071, A350073.

See also A069070, A336547, A349756, A364131.

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, Times @@ f2 @@@ f]; Array[a, 60] (* Amiram Eldar, Dec 12 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };
A350072(n) = A349162(n^2);

a(n) = A349162(n^2).

a(n) = A065764(n) / A350071(n).

A065827 Sum of squares of divisors of square numbers.

Original entry on oeis.org

1, 21, 91, 341, 651, 1911, 2451, 5461, 7381, 13671, 14763, 31031, 28731, 51471, 59241, 87381, 83811, 155001, 130683, 221991, 223041, 310023, 280371, 496951, 406901, 603351, 597871, 835791, 708123, 1244061, 924483, 1398101, 1343433, 1760031, 1595601, 2516921

Offset: 1

Vladeta Jovovic, Dec 06 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Harry J. Smith)

Cf. A001157, A065764.

A065827 := proc(n) numtheory[sigma][2](n^2) ; end proc:
seq(A065827(n),n=1..20) ; # R. J. Mathar, Apr 01 2011

DivisorSigma[2,#]&/@(Range[40]^2) (* Harvey P. Dale, May 18 2011 *)
f[p_, e_] := (p^(4*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

{ for (n=1, 500, a=sigma(n^2, 2); write("b065827.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 01 2009

[sigma(n^2,2)for n in range(1,34)] # Zerinvary Lajos, Jun 13 2009

Multiplicative with a(p^e) = (p^(4*e+2)-1)/(p^2-1).
a(n) = A001157(n^2). - R. J. Mathar, Mar 31 2011
Dirichlet g.f. zeta(s)*zeta(s-2)*zeta(s-4)/zeta(2s-4). Dirichlet convolution of A001159 by the arithmetic function with terms n^2*A008966(n). - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ 189 * Zeta(3) * Zeta(5) * n^5 / Pi^6. - Vaclav Kotesovec, Feb 01 2019
Sum_{k>=1} 1/a(k) = 1.06464520174524878494847955427968776606386158167258511428260450690334042955... - Vaclav Kotesovec, Sep 20 2020

A202994 a(n) = sigma(n^4).

Original entry on oeis.org

1, 31, 121, 511, 781, 3751, 2801, 8191, 9841, 24211, 16105, 61831, 30941, 86831, 94501, 131071, 88741, 305071, 137561, 399091, 338921, 499255, 292561, 991111, 488281, 959171, 797161, 1431311, 732541, 2929531, 954305, 2097151, 1948705, 2750971, 2187581

Offset: 1

Paul D. Hanna, Dec 27 2011

Here sigma(n^4) denotes the sums of divisors of n^4.

			L.g.f.: L(x) = x + 31*x^2/2 + 121*x^3/3 + 511*x^4/4 + 781*x^5/5 + 3751*x^6/6 +...
where exp(L(x)) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...+ A202993(n)*x^n +...

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000203 (sigma), A000583, A013663, A065764, A175926, A202993.

DivisorSigma[1,Range[40]^4] (* Harvey P. Dale, Jan 29 2012 *)
f[p_, e_] := (p^(4*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

PARI
```
{a(n)=sigma(n^4)}
```

from math import prod
from sympy import factorint
def A202994(n): return prod((p**((e<<2)+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

a(11*n) == 0 (mod 5) iff gcd(n,11) = 1.
Logarithmic derivative of A202993.
Multiplicative with a(p^e) = (p^(4*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018
a(n) = A000203(A000583(n)). - Michel Marcus, Sep 10 2020
Sum_{k>=1} 1/a(k) = 1.04483665108279017775482622699860068916340892303889072390102812885655694752... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.3840585791... . - Amiram Eldar, Nov 05 2022

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A263317 Least prime p > n such that the numbers sigma(k^2)/k^2 (k = 1,...,n) are pairwise incongruent modulo p, where sigma(m) is the sum of the divisors of m.

Original entry on oeis.org

2, 5, 5, 7, 7, 29, 37, 37, 37, 37, 37, 43, 43, 43, 53, 79, 101, 101, 101, 101, 101, 101, 101, 101, 131, 131, 131, 131, 131, 131, 131, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 283, 317, 389, 389, 389, 389, 389, 389, 389, 389, 389

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: a(n) exists for any n > 0, and a(n) < n^2 for all n > 2.
This implies that all the rational numbers sigma(n^2)/n^2 = Sum_{d|n^2} 1/d (n = 1,2,3,...) are pairwise distinct. We have verified that the numbers sigma(n^2)/n^2 (n = 1..10^5) are indeed pairwise distinct, and noted that sigma(26334^2)/26334^2 - sigma(6^2)/6^2 = 127/36 - 91/36 = 1.
We guess that for each k = 2,3,... all the numbers sigma(n^k)/n^k = Sum_{d|n^k} 1/d (n = 1,2,3,...) are pairwise distinct. See also A001157 for a similar conjecture.

			a(1) = 2 since 2 is the least prime greater than sigma(1^2)/1^2 = 1.
a(2) = 5 since sigma(1^2)/1^2 = 1 and sigma(2^2)/2^2 = 7/4 are incongruent modulo the prime 5 > 2, but 1 is congruent to 7/4 modulo the prime 3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Cf. A000203, A000290, A001157, A065764.

rMod[m_,n_]:=rMod[m,n]=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
f[n_]:=f[n]=DivisorSigma[1,n^2]/n^2
Le[n_,m_]:=Le[m,n]=Length[Union[Table[rMod[f[k],Prime[m]],{k,1,n}]]]
Do[n=1;m=1;Label[aa];If[m>PrimePi[n]&&Le[n,m]==n,Goto[bb],m=m+1;Goto[aa]];Label[bb];Print[n," ",Prime[m]];If[n<60,n=n+1;Goto[aa]]]

Definition corrected by Omar E. Pol, Oct 24 2015

A346865 Sum of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 12, 24, 56, 78, 144, 112, 360, 234, 360, 384, 672, 434, 960, 720, 992, 864, 1872, 760, 2352, 1344, 1584, 1872, 2880, 1767, 3024, 2160, 4032, 2400, 4320, 1984, 6552, 4032, 3672, 4608, 6552, 2812, 7440, 5376, 7200, 5082, 8064, 4752, 10080, 7020, 8064, 6144

Offset: 1

Omar E. Pol, Aug 17 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak.
So knowing this characteristic shape we can know if a number is an hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.
Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.

			a(3) = 24 because the sum of divisors of the third hexagonal number (i.e., 15) is 1 + 3 + 5 + 15 = 24.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a valley and the largest Dyck path has a peak as shown below.
Illustration of initial terms:
-------------------------------------------------------------------------
  n  H(n)  a(n)  Diagram
-------------------------------------------------------------------------
                 _         _                 _                         _
  1    1    1   |_|       | |               | |                       | |
                          | |               | |                       | |
                       _ _| |               | |                       | |
                      |    _|               | |                       | |
                 _ _ _|  _|                 | |                       | |
  2    6   12   |_ _ _ _|                   | |                       | |
                                            | |                       | |
                                       _ _ _|_|                       | |
                                   _ _| |                             | |
                                  |    _|                             | |
                                 _|  _|                               | |
                                |_ _|                                 | |
                                |                                     | |
                 _ _ _ _ _ _ _ _|                            _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|                           |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|
                                             _ _ _| |
                                            |  _ _ _|
                                            | |
                                            | |
                                            | |
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column H gives the nonzero hexagonal numbers (A000384).
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 8 + 8 + 8 = 24, so a(3) = 24.
For more information see A237593.

Bisection of A074285.
Cf. A000203, A000384, A237591, A237593, A245092, A262626, A346864.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

a[n_] := DivisorSigma[1, n*(2*n - 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)

a(n) = sigma(n*(2*n-1)); \\ Michel Marcus, Aug 18 2021

from sympy import divisors
def a(n): return sum(divisors(n*(2*n - 1)))
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Aug 20 2021

a(n) = A000203(A000384(n)).

Sum_{k=1..n} a(k) ~ 4*n^3/3. - Vaclav Kotesovec, Aug 18 2021

A346866 Sum of divisors of the n-th second hexagonal number.

Original entry on oeis.org

4, 18, 32, 91, 72, 168, 192, 270, 260, 576, 288, 868, 560, 720, 768, 1488, 864, 1482, 1120, 1764, 1408, 2808, 1152, 3420, 2232, 2268, 2880, 4480, 1800, 4464, 3328, 5292, 3264, 5184, 3456, 6734, 4712, 5760, 4480, 10890, 3528, 10368, 5280, 7560, 8736, 9216, 5760, 12152

Offset: 1

Omar E. Pol, Aug 17 2021

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.

			a(3) = 32 because the sum of divisors of the third second hexagonal number (i.e., 21) is 1 + 3 + 7 + 21 = 32.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a peak and the largest Dyck path has a valley as shown below.
Illustration of initial terms:
---------------------------------------------------------------------------------------
  n  h(n)  a(n)  Diagram
---------------------------------------------------------------------------------------
                    _             _                     _                            _
                   | |           | |                   | |                          | |
                _ _|_|           | |                   | |                          | |
  1    3    4  |_ _|             | |                   | |                          | |
                                 | |                   | |                          | |
                              _ _| |                   | |                          | |
                             |  _ _|                   | |                          | |
                          _ _|_|                       | |                          | |
                         |  _|                         | |                          | |
                _ _ _ _ _| |                           | |                          | |
  2   10   18  |_ _ _ _ _ _|                           | |                          | |
                                                _ _ _ _|_|                          | |
                                               | |                                  | |
                                              _| |                                  | |
                                             |  _|                                  | |
                                          _ _|_|                                    | |
                                      _ _|  _|                                      | |
                                     |_ _ _|                                        | |
                                     |                                 _ _ _ _ _ _ _| |
                                     |                                |    _ _ _ _ _ _|
                _ _ _ _ _ _ _ _ _ _ _|                                |   |
  3   21   32  |_ _ _ _ _ _ _ _ _ _ _|                             _ _|   |
                                                                  |       |
                                                                 _|    _ _|
                                                                |     |
                                                             _ _|    _|
                                                         _ _|      _|
                                                        |        _|
                                                   _ _ _|    _ _|
                                                  |         |
                                                  |  _ _ _ _|
                                                  | |
                                                  | |
                                                  | |
                                                  | |
               _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   36   91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column h gives the n-th second hexagonal number (A014105).
The widths of the main diagonal of the diagrams are [0, 0, 0, 1] respectively.
a(n) is the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 11 + 5 + 5 + 11 = 32, so a(3) = 32.
For n = 4 there is only one region (or part) of size 91 in the fourth diagram so a(4) = 91.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Bisection of A074285.
Cf. A000203, A000384, A014105, A237591, A237593, A245092, A262626, A346864.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors), A347155 (of nontriangular numbers).

a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)

a(n) = sigma(n*(2*n + 1)); \\ Michel Marcus, Aug 18 2021

a(n) = A000203(A014105(n)).

Sum_{k=1..n} a(k) ~ 4*n^3/3. - Amiram Eldar, Dec 31 2024

A074216 Squares satisfying sigma(n)==0 (mod 3).

Original entry on oeis.org

49, 169, 196, 361, 441, 676, 784, 961, 1225, 1369, 1444, 1521, 1764, 1849, 2704, 3136, 3249, 3721, 3844, 3969, 4225, 4489, 4900, 5329, 5476, 5776, 5929, 6084, 6241, 7056, 7396, 8281, 8649, 9025, 9409, 10609, 10816, 11025, 11881, 12321, 12544

Offset: 1

Benoit Cloitre, Sep 17 2002

nonn

Seems to contain all numbers of form k^2*p^2 where p are primes in A002476, k is not congruent to p and >=1.

Squares in A067051. - Michel Marcus, Dec 26 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A067051, A065764.

[n: n in [1..14161]|IsSquare(n) and DivisorSigma(1,n) mod 3 eq 0 ]; // Marius A. Burtea, Aug 17 2019

with(numtheory); A074216:=n->`if`(1-ceil(sigma(n^2)/3)+floor(sigma(n^2)/3)=1,n^2,NULL); seq(A074216(n), n=1..200); # Wesley Ivan Hurt, Dec 06 2013

Select[Range[150]^2,Divisible[DivisorSigma[1,#],3]&] (* Harvey P. Dale, Jul 10 2012 *)

isok(n) = issquare(n) && !(sigma(n) % 3); \\ Michel Marcus, Aug 17 2019

Conjecture: a(n) = A072864(n)^2. - R. J. Mathar, May 19 2020

A074964 Numbers k such that Max ( sigma(x*y) : 1 <= x <= k, 1 <= y <= k ) = sigma(k^2).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 18, 24, 60

Offset: 1

Benoit Cloitre, Oct 05 2002

Sequence is probably finite.
The next term in the sequence, if it exists, is larger than 40000. - Stewart Gordon, Sep 27 2011
Conjecture: subsequence of A066522, implying finiteness. - Reinhard Zumkeller, Nov 14 2011

Cf. A000203, A066522, A065764, A074963.

a074964 n = a074964_list !! (n-1)
a074964_list = filter (\x -> a074963 x == a065764 x) [1..]
-- Reinhard Zumkeller, Nov 14 2011

with(numtheory): s := proc(n) option remember: return sigma(n): end: a:= proc(n) option remember: if(n=0)then return 0: fi: return max(a(n-1),seq(s(x*n),x=1..n)): end: for n from 1 to 100 do if(a(n)=s(n^2))then printf("%d, ", n): end: od: # Nathaniel Johnston, Sep 26 2011

isok(k) = vecmax(setbinop((x,y)->sigma(x*y), [1..k])) == sigma(k^2); \\ Michel Marcus, Feb 03 2022

A074963(a(n)) = A065764(a(n)). - Reinhard Zumkeller, Nov 14 2011

A081325 a(n) = sigma(n^2) modulo 4.

Original entry on oeis.org

1, 3, 1, 3, 3, 3, 1, 3, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 3, 3, 3, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 3, 3, 3, 1, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 1, 1, 1, 3

Offset: 1

Benoit Cloitre, Apr 20 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A010873, A065764, A081334, A081339, A081354.

a[n_] := Mod[DivisorSigma[1, n^2], 4]; Array[a, 100] (* Amiram Eldar, May 03 2025 *)

a(n) = sigma(n^2) % 4; \\ Amiram Eldar, May 03 2025

From Amiram Eldar, May 03 2025: (Start)
a(n) = A010873(A065764(n)).
a(A081339(n)) = 1.
a(A081354(n)) = 3. (End)

A203556 a(n) = sigma(n^5).

Original entry on oeis.org

1, 63, 364, 2047, 3906, 22932, 19608, 65535, 88573, 246078, 177156, 745108, 402234, 1235304, 1421784, 2097151, 1508598, 5580099, 2613660, 7995582, 7137312, 11160828, 6728904, 23854740, 12207031, 25340742, 21523360, 40137576, 21243690, 89572392, 29583456, 67108863

Offset: 1

Paul D. Hanna, Jan 03 2012

a(n) modulo 6 begins: [1,3,4,1,0,0,0,3,1,0,0,4,0,0,0,1,0,3,0,0,0,0,0,0,1,0,...], in which positions of nonzero residues seem related to squares.

			L.g.f.: L(x) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...
where the g.f. of A203557 begins:
exp(L(x)) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A203557 (exp), A000203 (sigma), A000584, A013664.

Variants: A065764, A175926, A202994.

f[p_, e_] := (p^(5*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 09 2020 *)
DivisorSigma[1,Range[40]^5] (* Harvey P. Dale, Dec 05 2021 *)

PARI
```
a(n) = sigma(n^5)
```

Logarithmic derivative of A203557.
Multiplicative with a(p^e) = (p^(5*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018
From Amiram Eldar, Nov 05 2022: (Start)
a(n) = A000203(A000584(n)) = A000203(n^5).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3220880186... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

cp's OEIS Frontend

A350072 a(n) = sigma(n^2) / gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A065827 Sum of squares of divisors of square numbers.

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A202994 a(n) = sigma(n^4).

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A263317 Least prime p > n such that the numbers sigma(k^2)/k^2 (k = 1,...,n) are pairwise incongruent modulo p, where sigma(m) is the sum of the divisors of m.

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A346865 Sum of divisors of the n-th hexagonal number.

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A346866 Sum of divisors of the n-th second hexagonal number.

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A074216 Squares satisfying sigma(n)==0 (mod 3).

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