A065764 -id:A065764 - cp's OEIS frontend

A232354 Numbers k that divide sigma(k^2) where sigma is the sum of divisors function (A000203).

1, 39, 793, 2379, 7137, 13167, 76921, 78507, 230763, 238887, 549549, 692289, 863577, 1491633, 1672209, 2076867, 4317885, 7615179, 8329831, 10441431, 23402223, 24989493, 37776123, 53306253, 53695813, 55871145, 74968479, 83766969, 133854435, 144688401, 161087439, 189437391

Offset: 1

Alex Ratushnyak, Nov 22 2013

nonn

Squarefree terms are: 1, 39, 793, 2379, 76921, 230763, 8329831, 24989493, 53695813, 161087439, ... Quotients are: 1, 61, 873, 3783, 11737, 26543, 85563, 141911, 370773, 417263, 1155561, ... - Michel Marcus, Nov 23 2013

Many terms are also in sequence A069520, cf. A232067 for the intersection of these two sequences. - M. F. Hasler, Nov 24 2013

Donovan Johnson, Table of n, a(n) for n = 1..200
Jose Arnaldo Bebita Dris, A new approach to odd perfect numbers via GCDs, arXiv:2202.08116 [math.NT], 2022.

Cf. A000203, A007691, A065764, A069520, A227302, A232067, A232703, A232704.

Select[Range[10^5], Divisible[DivisorSigma[1, #^2], #] &] (* Alonso del Arte, Dec 06 2013 *)

isok(n) = (sigma(n^2) % n) == 0; \\ Michel Marcus, Nov 23 2013

A065764(a(n)) mod a(n) = 0.

A346867 Sum of divisors of the numbers that have middle divisors.

Original entry on oeis.org

1, 3, 7, 12, 15, 13, 28, 24, 31, 39, 42, 60, 31, 56, 72, 63, 48, 91, 90, 96, 78, 124, 57, 93, 120, 120, 168, 104, 127, 144, 144, 195, 96, 186, 121, 224, 180, 234, 112, 252, 171, 156, 217, 210, 280, 216, 248, 182, 360, 133, 312, 255, 252, 336, 240, 336, 168, 403, 372, 234

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is >= 1.
Also the width on the main diagonal equals the number of middle divisors.
So knowing this characteristic shape we can know if a number has middle divisors (or not) and the number of them just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.

			a(4) = 12 because the sum of divisors of the fourth number that has middle divisors (i.e., 6) is 1 + 2 + 3 + 6 = 12.
On the other hand we can see that in the main diagonal of every diagram the width is >= 1 as shown below.
Illustration of initial terms:
m(n) = A071562(n).
.
   n   m(n) a(n)   Diagram
.                  _ _   _   _   _ _     _     _ _   _   _       _
   1    1    1    |_| | | | | | | | |   | |   | | | | | | |     | |
   2    2    3    |_ _|_| | | | | | |   | |   | | | | | | |     | |
                   _ _|  _|_| | | | |   | |   | | | | | | |     | |
   3    4    7    |_ _ _|    _|_| | |   | |   | | | | | | |     | |
                   _ _ _|  _|  _ _|_|   | |   | | | | | | |     | |
   4    6   12    |_ _ _ _|  _| |  _ _ _| |   | | | | | | |     | |
                   _ _ _ _| |_ _|_|    _ _|   | | | | | | |     | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | |     | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | |     | |
                              |  _ _|    _| |    _ _ _|_| |     | |
                   _ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|     | |
   7   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| |    _ _ _ _ _| |
                                  |  _ _|  _|    _|   |    _ _ _ _|
                   _ _ _ _ _ _ _ _| |     |     |  _ _|   |
   8   15   24    |_ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |
   9   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|
                   _ _ _ _ _ _ _ _ _| | |     |      _|
  10   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|
                   _ _ _ _ _ _ _ _ _ _| | |       |
  11   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
                                          | |
                                          | |
                   _ _ _ _ _ _ _ _ _ _ _ _| |
  12   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
The n-th diagram has the property that at least it shares a vertex with the (n+1)-st diagram.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A240542, A245092, A249351, A262626, A281007, A299777, 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), A346866 (of second hexagonal numbers), A346868 (of numbers with no middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], Plus @@ d, 0]]; Select[Array[s, 150], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 ; \\ A071562
apply(sigma, select(is, [1..200])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071562(n)).

A378999 Number of trailing 1-bits in the binary representation of sigma(n^2).

Original entry on oeis.org

1, 3, 1, 5, 5, 2, 1, 7, 1, 1, 1, 2, 3, 4, 2, 9, 2, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 3, 1, 1, 11, 1, 1, 3, 3, 7, 2, 2, 1, 2, 2, 1, 2, 3, 5, 1, 2, 1, 2, 3, 1, 4, 2, 2, 3, 1, 1, 1, 1, 3, 3, 1, 13, 1, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 6, 3, 1, 3, 2, 2, 2, 3, 1, 2, 6, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Dec 16 2024

nonn

Cf. A000203, A000290, A007814, A065764, A378998

IntegerExponent[DivisorSigma[1, Range[100]^2] + 1, 2] (* Paolo Xausa, Dec 19 2024 *)

PARI
```
A378999(n) = valuation(sigma(n^2)+1,2);
```

a(n) = A378998(A000290(n)).

a(n) = A007814(1+A065764(n)). [the 2-adic valuation of 1+sigma(n^2)]

A074465 a(n) = gcd(n^2, sigma(n^2), phi(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 39, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 11, 1, 1, 1, 1

Offset: 1

Labos Elemer, Aug 23 2002

nonn

a(n) is odd because sigma(n^2) is odd;.

			For n=14: gcd(196,399,84) = 7 = a(14).

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

Cf. A000203, A002618, A065764, A074389, A074466.

Table[Apply[GCD, {w^2, DivisorSigma[1, w^2], EulerPhi[w^2]}], {w, 1, 128}]

A074465(n) = gcd([n^2, sigma(n^2), eulerphi(n^2)]); \\ Antti Karttunen, Sep 07 2018

a(n) = A074389(n^2).

A081339 Numbers n such that sigma(n^2) modulo 4 = 1.

Original entry on oeis.org

1, 3, 7, 9, 10, 11, 19, 20, 21, 23, 25, 26, 27, 30, 31, 33, 34, 40, 43, 47, 49, 52, 57, 58, 59, 60, 63, 65, 67, 68, 69, 70, 71, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 90, 93, 99, 102, 103, 104, 106, 107, 110, 116, 120, 121, 122, 127, 129, 131, 133, 136, 139, 140, 141, 145

Offset: 1

Benoit Cloitre, Apr 20 2003

nonn

Numbers n such that the sum of exponents of primes == 1 (mod 4) in the prime factorization of n is not congruent to n mod 2. - Robert Israel, Jan 22 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A065764, A081325.

Contains A004614.

filter:= proc(n) local F, t;
  F:= select(t -> t[1] mod 4 = 1, ifactors(n)[2]);
  (add(t[2],t=F) - n) mod 2 = 1;
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 22 2017

Select[Range[150],Mod[DivisorSigma[1,#^2],4]==1&] (* Harvey P. Dale, Apr 07 2012 *)

isok(n) = (sigma(n^2) % 4) == 1; \\ Michel Marcus, Jan 22 2017

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

A195735 a(n) = 2*sigma(n^2) - sigma(n)^2.

Original entry on oeis.org

1, 5, 10, 13, 26, 38, 50, 29, 73, 110, 122, 22, 170, 222, 230, 61, 290, 173, 362, 158, 458, 566, 530, -298, 601, 798, 586, 398, 842, 458, 962, 125, 1154, 1382, 1230, -779, 1370, 1734, 1622, -226, 1682, 1158, 1850, 1190, 1418, 2558, 2210, -2090, 2353, 2285, 2798, 1742, 2810, 902, 3062, 78, 3506, 4094, 3482, -3238

Offset: 1

Paul D. Hanna, Sep 22 2011

sign

			L.g.f.: L(x) = x + 5*x^2/2 + 10*x^3/3 + 13*x^4/4 + 26*x^5/5 + 38*x^6/6 +...
where exp(L(x)) = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 22*x^5 + 40*x^6 + 72*x^7 + 123*x^8 + 215*x^9 + 363*x^10 +...+ A195734(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A195734 (exp), A067807, A065764, A000203, A002117.

with(numtheory); A195735:=n->2*sigma(n^2) - sigma(n)^2; seq(A195735(n), n=1..100); # Wesley Ivan Hurt, Mar 04 2014

Table[2 DivisorSigma[1, n^2] - DivisorSigma[1, n]^2, {n, 100}] (* Wesley Ivan Hurt, Mar 04 2014 *)

PARI
```
{a(n)=2*sigma(n^2) - sigma(n)^2}
```

a(n) < 0 for n found in A067807.
Equals the logarithmic derivative of A195734.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (10/Pi^2-5/6)*zeta(3) = 0.216224196369... . - Amiram Eldar, Mar 17 2024

A224339 Absolute difference between sum of odd divisors of n^2 and sum of even divisors of n^2.

Original entry on oeis.org

1, 5, 13, 29, 31, 65, 57, 125, 121, 155, 133, 377, 183, 285, 403, 509, 307, 605, 381, 899, 741, 665, 553, 1625, 781, 915, 1093, 1653, 871, 2015, 993, 2045, 1729, 1535, 1767, 3509, 1407, 1905, 2379, 3875, 1723, 3705, 1893, 3857, 3751, 2765, 2257, 6617, 2801, 3905, 3991, 5307

Offset: 1

Paul D. Hanna, Apr 03 2013

Multiplicative because A113184 is.

Logarithmic derivative of A224340.

			L.g.f.: L(x) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 31*x^5/5 + 65*x^6/6 + 57*x^7/7 + 125*x^8/8 + 121*x^9/9 + 155*x^10/10 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 64*x^6 + 120*x^7 + 236*x^8 + 434*x^9 + 805*x^10 +...+ A224340(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A065764, A113184, A224340.

dif[n_]:=Module[{divs=Divisors[n^2],od,ev},od=Total[Select[divs,OddQ]];ev=Total[Select[divs,EvenQ]];Abs[od-ev]]; Array[dif,60] (* Harvey P. Dale, Jul 16 2015 *)
f[p_, e_] := If[p == 2, 2^(2*e + 1) - 3, (p^(2*e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 01 2022 *)

{a(n)=if(n<1, 0, (-1)^n*sumdiv(n^2, d, (-1)^d*d))}
for(n=1,64,print1(a(n),", "))

a(n) = if(n%2, sigma(n^2), 4*sigma(n^2/2) - sigma(n^2)) \\ Andrew Howroyd, Jul 28 2018

a(n) = (-1)^n * Sum_{d|n^2} (-1)^d * d.
a(n) = A113184(n^2).
a(n) = sigma(n^2) for odd n; a(n) = 4*sigma(n^2/2) - sigma(n^2) for even n. - Andrew Howroyd, Jul 28 2018
Multiplicative with a(p^e) = 2^(2*e+1)-3 if p=2, and (p^(2*e+1)-1)/(p-1) otherwise. - Amiram Eldar, Jul 01 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = (9*zeta(3))/(2*Pi^2) = 0.548072... . - Amiram Eldar, Oct 13 2022

A232703 Numbers k that divide sigma(k) + sigma(k^2).

Original entry on oeis.org

1, 2, 185, 113125, 535350201, 71373089217, 82225037115, 156231750533, 268821356379

Offset: 1

Alex Ratushnyak, Nov 28 2013

a(10) > 10^12. - Giovanni Resta, Jun 10 2016

Cf. A000203 (sigma(n) = sum of divisors of n).

Cf. A232547, A065764, A247222.

Select[Range[150000], Mod[Plus @@ DivisorSigma[1, {#, #^2}], #] == 0 &] (* Giovanni Resta, Jun 10 2016 *)

a(6)-a(9) from Giovanni Resta, Jun 10 2016

A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

Original entry on oeis.org

0, 6, 16, 42, 54, 132, 120, 270, 286, 450, 360, 952, 546, 1056, 1152, 1674, 1098, 2574, 1440, 3276, 2720, 3312, 2376, 6300, 3534, 5124, 5160, 7672, 4380, 11088, 5184, 10836, 8688, 10314, 9600, 19110, 8322, 13680, 13440, 22590, 11046, 26304, 12452, 24780, 23868, 22968, 15792, 42408, 20349, 34131

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers sigma(n)*pi(n*(n+1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers n*sigma(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct, and all the numbers sigma(n^2)*pi(n^2) (n = 1,2,3,...) are also pairwise distinct.
(iv) All the numbers n*phi(n)*sigma(n^2) = phi(n^2)*sigma(n^2) (n = 1,2,3,...) are pairwise distinct, where phi(.) is Euler's totient function.
We have verified that the terms a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263319 for a similar conjecture.

			a(1) = 0 since sigma(1)*pi(1^2) = 1*0 = 0.
a(2) = 6 since sigma(2)*pi(2^2) = 3*2 = 6.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A000290, A000720, A002618, A038107, A065764, A263317, A263319.

[#PrimesUpTo(n^2)*SumOfDivisors(n): n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=DivisorSigma[1,n]*PrimePi[n^2]
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = sigma(n)*primepi(n^2); \\ Michel Marcus, Oct 15 2015

cp's OEIS Frontend

A232354 Numbers k that divide sigma(k^2) where sigma is the sum of divisors function (A000203).

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A346867 Sum of divisors of the numbers that have middle divisors.

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A378999 Number of trailing 1-bits in the binary representation of sigma(n^2).

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A074465 a(n) = gcd(n^2, sigma(n^2), phi(n^2)).

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A081339 Numbers n such that sigma(n^2) modulo 4 = 1.

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

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A195735 a(n) = 2*sigma(n^2) - sigma(n)^2.

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A224339 Absolute difference between sum of odd divisors of n^2 and sum of even divisors of n^2.