A113184 -id:A113184 - cp's OEIS frontend

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

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

A033832 Sum of odd divisors of n < sqrt(n) = sum of even divisors of n < sqrt(n).

Original entry on oeis.org

1, 40, 100, 208, 928, 1044, 3904, 10692, 17444, 29524, 36652, 45980, 87604, 91044, 136808, 158652, 161564, 171028, 187068, 218652, 230044, 260608, 287868, 406812, 438124, 450492, 583110, 665684, 719550, 731850, 736648, 865444, 1045504

Offset: 1

Naohiro Nomoto

All terms except first one appear to be even. - Michel Marcus, Jul 15 2013

Amiram Eldar, Table of n, a(n) for n = 1..500
kaleidoscopic numbers [broken link]

Cf. A070039, A113184.

aQ[n_] := DivisorSum[n, # * (-1)^# &, # < Sqrt[n] & ] == 0; Select[Range[10^4], aQ] (* Amiram Eldar, Sep 23 2019 *)

isok(n) = {so = 0; se = 0; fordiv (n, d, if (d < sqrt(n), if (d % 2, so += d, se += d))); return (so == se);} \\ Michel Marcus, Jul 14 2013

Prepended a(1)=1, Michel Marcus, Jul 15 2013

A326238 Expansion of Sum_{k>=1} k * x^k * (1 - x^k) / (1 + x^k)^3.

Original entry on oeis.org

1, -2, 12, -20, 30, -24, 56, -104, 117, -60, 132, -240, 182, -112, 360, -464, 306, -234, 380, -600, 672, -264, 552, -1248, 775, -364, 1080, -1120, 870, -720, 992, -1952, 1584, -612, 1680, -2340, 1406, -760, 2184, -3120, 1722, -1344, 1892, -2640, 3510, -1104, 2256

Offset: 1

Ilya Gutkovskiy, Sep 10 2019

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

Cf. A002129, A064987, A113184, A143520, A185152, A326125.

nmax = 47; CoefficientList[Series[Sum[k x^k (1 - x^k)/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(d + 1) d, {d, Divisors[n]}], {n, 1, 47}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e*(3-2^(e+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n)={n*sumdiv(n, d, (-1)^(d + 1) * d)} \\ Andrew Howroyd, Sep 10 2019

G.f.: Sum_{k>=1} (-1)^(k + 1) * k^2 * x^k / (1 - x^k)^2.
a(n) = n * Sum_{d|n} (-1)^(d + 1) * d.
a(n) = n * A002129(n).
Multiplicative with a(2^e) = 2^e*(3-2^(e+1)), and a(p^e) = p^e*(p^(e+1)-1)/(p-1) if p > 2. - Amiram Eldar, Dec 05 2022
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(3-s)). - Amiram Eldar, Jan 07 2023

A143336 Expansion of K(k) * (2 * E(k) - K(k)) / (Pi/2)^2 in powers of q where E(k), K(k) are complete elliptic integrals and q = exp(-Pi * K(k') / K(k)).

Original entry on oeis.org

1, -8, -8, -32, -40, -48, -32, -64, -104, -104, -48, -96, -160, -112, -64, -192, -232, -144, -104, -160, -240, -256, -96, -192, -416, -248, -112, -320, -320, -240, -192, -256, -488, -384, -144, -384, -520, -304, -160, -448, -624, -336, -256, -352, -480, -624, -192, -384, -928, -456, -248, -576, -560, -432

Offset: 0

Michael Somos, Aug 09 2008

sign

			G.f. = 1 - 8*q - 8*q^2 - 32*q^3 - 40*q^4 - 48*q^5 - 32*q^6 - 64*q^7 - 104*q^8 + ...

Cf. A113184, A122858.

a[ n_] := If[ n < 1, Boole[n == 0], -(-1)^n 8 Sum[(-1)^d d, {d, Divisors @ n}]]; (* Michael Somos, Apr 07 2015 *)
a[ n_] := SeriesCoefficient[ With[{m = InverseEllipticNomeQ[ q]}, EllipticK[ m] (2 EllipticE[ m] - EllipticK[ m]) (2/Pi)^2], {q, 0, n}]; (* Michael Somos, Apr 07 2015 *)

{a(n) = if( n<1, n==0, -(-1)^n * 8 * sumdiv(n, d, (-1)^d * d))};

The generating function equals 0 when 2 * E(k) = K(k) at q = 0.1076539192... (A072558) the "One-Ninth" constant.
Expansion of (P(q) - 2 * P(q^2) + 4 * P(q^4)) / 3 in powers of q where P() is a Ramanujan Lambert series.
G.f.: 1 - 8 * Sum_{k>0} k * x^k / (1 - (-x)^k) = 1 + 8 * Sum_{k>0} (-x)^k / (1 + (-x)^k)^2.
a(n) = (-1)^n * A122858(n). a(n) = -8 * A113184(n) unless n=0.

A348583 Numbers k such that k | A002129(k).

Original entry on oeis.org

1, 60, 728, 6960, 60512, 97152, 728000, 1900080, 2184000, 4371840, 26522496, 843480000, 23009688000, 46352390400, 93155148800, 279465446400, 701869363200, 938948846080, 1099176108032, 2816846538240

Offset: 1

Amiram Eldar, Oct 24 2021

Equivalently, numbers k such that k | A113184(k).
The corresponding ratios A002129(k)/k are 1, -2, -2, -3, -2, -3, -3, -4, -4, -4, -4, -4, -4, -4, -3, -4, -4, -3, -2, -4, ...
If p is a Mersenne exponent (A000043), and the corresponding Mersenne prime (A000668) M_p = 2^p - 1 is in A005382 or A167917, i.e., 2*M_p - 1 is also a prime, then 2^p*(2^p-1)*(2^(p+1)-3) is a term. The corresponding known terms of this form are 60, 728, 60512, 1099176108032 and 288229001763749888.
If a term k is odd, then A002129(k) = A000203(k) and thus k is a multiply-perfect number. Therefore, the odd perfect numbers, if they exist, are terms of this sequence.

			60 is a term since A002129(60) = -120 is divisible by 60.

Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A002129, A005382, A113184, A167917.

f[p_, e_] := If[p == 2, 2^(e + 1)-3, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], Divisible[s[#], #] &]

a(20) from Martin Ehrenstein, Nov 06 2021

A348584 Numbers k such that k | A328258(k).

Original entry on oeis.org

1, 12, 56, 180, 992, 16256, 127400, 441000, 2646000, 67100672, 325458000, 2758909440, 17179738112, 274877382656

Offset: 1

Amiram Eldar, Oct 24 2021

The corresponding ratios A113184(k)/k are 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -1, -1, ...
If p is a Mersenne exponent (A000043), then 2^p*(2^p-1) (twice an even perfect number) is a term with ratio A328258(k)/k = -1.
If there exists an odd term k, then it is a unitary multiply-perfect number (A327158), since A328258(k) = A034448(k) for an odd k.

			12 is a term since A328258(12) = -12 is divisible by 12.

The unitary version of A348583.
A139256 is a subsequence.
Cf. A000043, A034448, A327158, A328258.

f[p_, e_] := 1 - (-1)^p*(p^e); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[3*10^6], Divisible[s[#], #] &]

cp's OEIS Frontend

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

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A033832 Sum of odd divisors of n < sqrt(n) = sum of even divisors of n < sqrt(n).

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A326238 Expansion of Sum_{k>=1} k * x^k * (1 - x^k) / (1 + x^k)^3.

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A143336 Expansion of K(k) * (2 * E(k) - K(k)) / (Pi/2)^2 in powers of q where E(k), K(k) are complete elliptic integrals and q = exp(-Pi * K(k') / K(k)).

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A348583 Numbers k such that k | A002129(k).

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A348584 Numbers k such that k | A328258(k).

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