A065765 -id:A065765 - cp's OEIS frontend

A065767 Intersection of A065764 and A065765: n such that x and y exist with sigma[x^2] = n = sigma[2*(y^2)].

399, 5187, 12369, 34671, 48279, 73017, 80199, 122493, 152019, 160797, 220647, 259749, 311619, 347529, 396207, 436107, 561393, 687477, 755307, 900543, 949221, 1042587, 1074801, 1142337, 1412859, 1496649, 1509417, 1592409, 1818243

Offset: 1

Labos Elemer, Nov 19 2001

nonn

			n = 399 = sigma[14^2] = sigma[2*(11^2)] = 1+2+4+7+14+28+49+98+196 = 1+2+11+22+121+242; also sigma[42.42] = sigma[2.33.33] = sigma[1764] = sigma[2378] = 5187.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A028982, A000203, A000290, A065764-A065768.

Intersection[Table[DivisorSigma[1, w^2], {w, 1, 10000}], Table[DivisorSigma[1, 2*(w^2)], {w, 1, 10000}]]

A065764 Sum of divisors of square numbers.

Original entry on oeis.org

1, 7, 13, 31, 31, 91, 57, 127, 121, 217, 133, 403, 183, 399, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 1281, 1093, 1767, 871, 2821, 993, 2047, 1729, 2149, 1767, 3751, 1407, 2667, 2379, 3937, 1723, 5187, 1893, 4123, 3751, 3871, 2257, 6643

Offset: 1

Labos Elemer, Nov 19 2001

Unlike A065765, the sums of divisors of squares give remainders r=1,3,5 modulo 6: sigma(4)==1, sigma(49)==3, sigma(2401)==5 (mod 6). See also A097022.
a(n) is also the number of ordered pairs of positive integers whose LCM is n, (see LeVeque). - Enrique Pérez Herrero, Aug 26 2013
Main diagonal of A319526. - Omar E. Pol, Sep 25 2018
Subsequence of primes is A023195 \ {3}; also, 31 is the only known prime to be twice in the data because 31 = sigma(16) = sigma(25) (see A119598 and Goormaghtigh conjecture link). - Bernard Schott, Jan 17 2021

W. J. LeVeque, Fundamentals of Number Theory, pp. 125 Problem 4, Dover NY 1996.

T. D. Noe, Table of n, a(n) for n=1..10000
Wikipedia, Goormaghtigh conjecture.

Cf. A000203, A000290, A028982, A319526.

a:=List([1..50],n->Sigma(n^2));; Print(a); # Muniru A Asiru, Jan 01 2019

[SumOfDivisors(n^2): n in [1..48]]; // Bruno Berselli, Apr 12 2011

with(numtheory): [sigma(n^2)$n=1..50]; # Muniru A Asiru, Jan 01 2019

Table[Plus@@Divisors[n^2], {n, 48}] (* Alonso del Arte, Feb 24 2012 *)
f[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

numlib::sigma(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

a(n) = sigma(n^2); \\ Harry J. Smith, Oct 30 2009

from math import prod
from sympy import factorint
def A065764(n): return prod((p**((e<<1)+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

[sigma(n^2,1)for n in range(1,49)] # Zerinvary Lajos, Jun 13 2009

a(n) = sigma(n^2) = A000203(A000290(n)).
Multiplicative with a(p^e) = (p^(2*e+1)-1)/(p-1). - Vladeta Jovovic, Dec 01 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/zeta(2*s-2), inverse Mobius transform of A000082. - R. J. Mathar, Mar 06 2011
Dirichlet convolution of A001157 by the absolute terms of A055615. Also the Dirichlet convolution of A048250 by A000290. - R. J. Mathar, Apr 12 2011
a(n) = Sum_{d|n} d*Psi(d), where Psi is A001615. - Enrique Pérez Herrero, Feb 25 2012
a(n) >= (n+1) * sigma(n) - n, where sigma is A000203, equality holds if n is in A000961. - Enrique Pérez Herrero, Apr 21 2012
Sum_{k=1..n} a(k) ~ 5*Zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, Jan 30 2019
Sum_{k>=1} 1/a(k) = 1.3947708738535614499846243600124612760835313454790187655653356563282177118... - Vaclav Kotesovec, Sep 20 2020

A065766 Sum of divisors of twice a square number, divided by three.

Original entry on oeis.org

1, 5, 13, 21, 31, 65, 57, 85, 121, 155, 133, 273, 183, 285, 403, 341, 307, 605, 381, 651, 741, 665, 553, 1105, 781, 915, 1093, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2541, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 3751, 2765, 2257, 4433, 2801

Offset: 1

Labos Elemer, Nov 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A028982, A000203, A000290, A002117, A065764, A065765, A065767.

List([1..50],n->Sigma(2*n^2))/3; # Muniru A Asiru, Dec 07 2018

[SumOfDivisors(2*n^2)/3: n in [1..60]]; // Vincenzo Librandi, Dec 07 2018

with(numtheory): [sigma(2*n^2)/3$n=1..50]; # Muniru A Asiru, Dec 07 2018

Array[DivisorSigma[1, 2 #^2]/3 &, 49] (* Michael De Vlieger, Dec 06 2018 *)

a(n) = { sigma(2*n^2)/3 } \\ Harry J. Smith, Oct 30 2009

from sympy import divisor_sigma
for n in range(1,50): print(divisor_sigma(2*n**2,1)/3) # Stefano Spezia, Dec 07 2018

Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(2*e+1)-1)/(p-1) for an odd prime p. - Vladeta Jovovic, Dec 01 2001
a(n) = sigma(2*n^2)/3 = A000203(2*A000290(n))/3 = A065765(n)/3.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 4*zeta(3)/Pi^2 = 0.487175... . - Amiram Eldar, Oct 28 2022
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/(zeta(2*s-2)*(1+2/2^s)). - Amiram Eldar, Feb 12 2023

A097022 a(n) = (sigma(2n^2)-3)/6.

Original entry on oeis.org

0, 2, 6, 10, 15, 32, 28, 42, 60, 77, 66, 136, 91, 142, 201, 170, 153, 302, 190, 325, 370, 332, 276, 552, 390, 457, 546, 598, 435, 1007, 496, 682, 864, 767, 883, 1270, 703, 952, 1189, 1317, 861, 1852, 946, 1396, 1875, 1382, 1128, 2216, 1400, 1952, 1995, 1921

Offset: 1

Labos Elemer, Aug 24 2004

nonn

See A065764, A065765 and A097022.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A065764, A065765, A000203, A001105, A002117, A072682, A074627, A074628, A074629, A074630, A067051.

Table[(DivisorSigma[1,2n^2]-3)/6,{n,60}] (* Harvey P. Dale, Sep 12 2022 *)

a(n) = (sigma(2*n^2) - 3)/6; \\ Michel Marcus, Dec 20 2013

a(n) = (A065765(n)-3)/6 = A000203(A001105(n) - 3)/6.

Sum_{k=1..n} a(k) ~ c * n^3, where c = 4*zeta(3)/Pi^2 = 0.243587... . - Amiram Eldar, Oct 28 2022

A081334 a(n) = sigma(2*n^2) modulo 4.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 20 2003

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

Cf. A000203, A010873, A065765, A081325.

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

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

a(n) = A010873(A065765(n)). - Amiram Eldar, May 03 2025

cp's OEIS Frontend

A065767 Intersection of A065764 and A065765: n such that x and y exist with sigma[x^2] = n = sigma[2*(y^2)].

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A065764 Sum of divisors of square numbers.

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A065766 Sum of divisors of twice a square number, divided by three.

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A097022 a(n) = (sigma(2n^2)-3)/6.

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A081334 a(n) = sigma(2*n^2) modulo 4.

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