A050469 -id:A050469 - cp's OEIS frontend

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

1, 1024, 59048, 1048576, 9765626, 60465152, 282475248, 1073741824, 3486725353, 10000001024, 25937424600, 61916315648, 137858491850, 289254653952, 576640684048, 1099511627776, 2015993900450, 3570406761472, 6131066257800, 10240001048576

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, this sequence, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^10 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(10*e+10) - s[p]^(e+1))/(p^10 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^10)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10) - A101455(p)^(e+1))/(p^10 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = beta(11) = 50521*Pi^11/14863564800 = 0.999994374973... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^10*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

Original entry on oeis.org

1, 2048, 177146, 4194304, 48828126, 362795008, 1977326742, 8589934592, 31380882463, 100000002048, 285311670610, 743004176384, 1792160394038, 4049565167616, 8649707208396, 17592186044416, 34271896307634, 64268047284224, 116490258898218

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for related sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, this sequence, A321836.

s[n_,r_] := DivisorSum[n, #^11 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(11*e+11) - s[p]^(e+1))/(p^11 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^11)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(11*e+11) - A101455(p)^(e+1))/(p^11 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = beta(12) = 0.99999812235..., and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^11*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 10, 12, 11, 12, 14, 14, 18, 16, 18, 20, 19, 24, 22, 22, 23, 24, 31, 28, 30, 28, 30, 36, 31, 32, 34, 36, 42, 40, 38, 38, 42, 48, 42, 44, 43, 44, 60, 46, 47, 48, 50, 62, 54, 56, 54, 60, 66, 56, 58, 60, 59, 72, 62, 62, 73, 64, 84, 68

Offset: 1

N. J. A. Sloane, Dec 23 1999

Not multiplicative: a(3)*a(7) <> a(21), for example.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001826, A002131, A050449, A050464, A050469, A222183.

Cf. A050461, A050462, A050463.

A050460 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050460(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#&]; Array[a, 70] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=sumdiv(n,d,if(n/d%4==1,d)) \\ Charles R Greathouse IV, Dec 04 2013

G.f.: Sum_{n>0} n*x^n/(1-x^(4*n)). - Vladeta Jovovic, Nov 14 2002
G.f.: Sum_{k>0} x^(4*k-3) / (1 - x^(4*k-3))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050464(n).
a(n) = A050469(n) + A050464(n).
a(n) = (A002131(n) + A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A222183. (End)

A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 4, 0, 2, 6, 0, 0, 6, 1, 0, 10, 2, 1, 8, 0, 0, 10, 4, 0, 12, 1, 0, 14, 0, 6, 12, 0, 2, 14, 0, 0, 20, 1, 4, 18, 2, 1, 16, 7, 0, 18, 0, 0, 20, 6, 8, 22, 0, 1, 24, 0, 2, 31, 0, 0, 28, 1, 0, 26, 12, 1, 24, 0, 0, 31, 4, 18, 28, 1, 0, 30, 0, 1

Offset: 1

N. J. A. Sloane, Dec 23 1999

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

Cf. A002131, A050460, A050469, A247037, A326400, A363900.

Cf. A050465, A050466, A050467.

a[n_] := DivisorSum[n, # &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

a(n)={sumdiv(n, d, d*(n/d%4==3))} \\ Andrew Howroyd, Sep 13 2019

G.f.: Sum_{k>=1} k*x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 13 2019
G.f.: Sum_{k>0} x^(4*k-1) / (1 - x^(4*k-1))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050460(n).
a(n) = A050460(n) - A050469(n).
a(n) = (A002131(n) - A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A247037. (End)

Offset changed to 1 by Ilya Gutkovskiy, Sep 13 2019

A364012 Expansion of Sum_{k>0} k * x^k / (1 + x^(3*k)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 8, 6, 9, 9, 11, 9, 14, 16, 15, 11, 17, 18, 20, 13, 24, 21, 23, 18, 26, 28, 27, 24, 29, 27, 32, 22, 33, 33, 40, 27, 38, 40, 42, 25, 41, 48, 44, 31, 45, 45, 47, 33, 57, 47, 51, 42, 53, 54, 56, 48, 60, 57, 59, 39, 62, 64, 72, 43, 70, 63, 68, 49, 69, 72, 71, 54, 74, 76, 78

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

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

Cf. A364011, A364013.

Cf. A050469, A364019, A364031.

a[n_] := -DivisorSum[n, (-1)^(n/#) * # &, Mod[n/#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)

a(n) = -sumdiv(n, d, (n/d%3==1)*(-1)^(n/d)*d);

G.f.: Sum_{k>0} (-1)^(k-1) * x^(3*k-2) / (1 - x^(3*k-2))^2.

a(n) = -Sum_{d|n, n/d==1 (mod 3)} (-1)^(n/d) * d.

A322084 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 4, 2, 1, 1, 8, 8, 4, 2, 1, 16, 26, 16, 6, 0, 1, 32, 80, 64, 26, 4, 0, 1, 64, 242, 256, 126, 32, 6, 1, 1, 128, 728, 1024, 626, 208, 48, 8, 1, 1, 256, 2186, 4096, 3126, 1280, 342, 64, 7, 2, 1, 512, 6560, 16384, 15626, 7744, 2400, 512, 73, 12, 0, 1, 1024, 19682, 65536, 78126, 46592, 16806, 4096, 703, 104, 10, 0

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  1,  2,   4,    8,    16,    32,  ...
  0,  2,   8,   26,    80,   242,  ...
  1,  4,  16,   64,   256,  1024,  ...
  2,  6,  26,  126,   626,  3126,  ...
  0,  4,  32,  208,  1280,  7744,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322083.

Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^(2 j)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, if(d%2, (-1)^((d-1)/2)*(n/d)^k))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^(2*j)).

A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

Original entry on oeis.org

1, 1, 3, 3, 4, 3, 8, 5, 9, 4, 10, 9, 14, 8, 12, 11, 16, 9, 20, 12, 24, 10, 22, 15, 21, 14, 27, 24, 28, 12, 32, 21, 30, 16, 32, 27, 38, 20, 42, 20, 40, 24, 44, 30, 36, 22, 46, 33, 57, 21, 48, 42, 52, 27, 40, 40, 60, 28, 58, 36, 62, 32, 72, 43, 56, 30, 68, 48, 66, 32

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia Rella, Resurgence, Stokes constants, and arithmetic functions in topological string theory, arXiv:2212.10606 [hep-th], 2022. See pages 21 - 23.

Cf. A000593, A002324, A050469, A078708, A326399, A326400.

Cf. A175646, A340577.

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 + x^k + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 70}]
f[p_, e_] := Which[p == 3, p^e, Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 3] == 2, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 3, 3^f[i,2], if(f[i,1]%3 == 1, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1), (f[i,1]^(f[i,2]+1) + (-1)^f[i,2])/(f[i,1] + 1))));} \\ Amiram Eldar, Nov 06 2022

a(n) = Sum_{d|n, n/d==1 (mod 3)} d - Sum_{d|n, n/d==2 (mod 3)} d.
a(n) = A326399(n) - A326400(n).
Multiplicative with a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1 (mod 3), and (p^(e+1) + (-1)^e)/(p + 1) if p == 2 (mod 3). - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) * Product_{primes p == 2 (mod 3)} 1/(1 + 1/p^2) = (1/2) * A175646 * (2*Pi^2/27)/A340577 = 0.3906512064... . - Amiram Eldar, Nov 06 2022

A327122 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 + x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 3, 7, 7, 9, 7, 15, 10, 21, 11, 21, 15, 21, 21, 31, 19, 30, 19, 49, 21, 33, 23, 45, 38, 45, 30, 49, 31, 63, 31, 63, 33, 57, 49, 70, 39, 57, 45, 105, 43, 63, 43, 77, 70, 69, 47, 93, 50, 114, 57, 105, 55, 90, 77, 105, 57, 93, 59, 147, 63, 93, 70, 127, 105

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

Inverse Moebius transform of A050469.

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

Cf. A000203, A050469, A101455, A167181 (fixed points), A262209, A288417, A327123.

Cf. A072691, A175647, A243381.

nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
A050469[n_] := DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 4]] &] - DivisorSum[n, # &, MemberQ[{3}, Mod[n/#, 4]] &]; a[n_] := DivisorSum[n, A050469[#] &]; Table[a[n], {n, 1, 65}]
f[p_, e_] := If[Mod[p, 4] == 1, (p^(e+2)-(e+2)*p+e+1)/(p-1)^2, (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1))]; f[2, e_] := 2^(e+1)-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Aug 28 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p == 2, 2^(e+1)-1, if(p%4 == 1, (p^(e+2)-(e+2)*p+e+1)/(p-1)^2, (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1))))); } \\ Amiram Eldar, Aug 28 2023

a(n) = Sum_{d|n} A050469(d).
From Amiram Eldar, Aug 28 2023: (Start)
Multiplicative with a(2^e) = 2^(e+1)-1, and if p is an odd prime a(p^e) = (p^(e+2)-(e+2)*p+e+1)/(p-1)^2 if p == 1 (mod 4) and (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1)) otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 * (A175647/A243381) = 0.753351504961... . (End)

A327123 Expansion of Sum_{k>=1} phi(k) * x^k / (1 + x^(2*k)), where phi = A000010.

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 5, 4, 5, 5, 9, 2, 13, 5, 5, 8, 17, 5, 17, 10, 5, 9, 21, 4, 25, 13, 13, 10, 29, 5, 29, 16, 9, 17, 25, 10, 37, 17, 13, 20, 41, 5, 41, 18, 25, 21, 45, 8, 37, 25, 17, 26, 53, 13, 45, 20, 17, 29, 57, 10, 61, 29, 25, 32, 65, 9, 65, 34, 21

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

Moebius transform of A050469.

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

Cf. A000010, A004613 (fixed points), A006752, A008683, A026741, A050469, A101455, A193356, A327122.

nmax = 69; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
A050469[n_] := DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 4]] &] - DivisorSum[n, # &, MemberQ[{3}, Mod[n/#, 4]] &]; a[n_] := DivisorSum[n, MoebiusMu[n/#] A050469[#] &]; Table[a[n], {n, 1, 69}]
f[p_, e_] := If[Mod[p, 4] == 1, p^e, (p^(e+1) - p^e + 2*(-1)^e)/(p+1)]; f[2, e_] := 2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Aug 28 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p == 2, 2^(e-1), if(p%4 == 1, p^e, (p^(e+1) - p^e + 2*(-1)^e)/(p+1)))); } \\ Amiram Eldar, Aug 28 2023

a(n) = Sum_{d|n} mu(n/d) * A050469(d).
From Amiram Eldar, Aug 28 2023: (Start)
Multiplicative with a(2^e) = 2^(e-1), and if p is an odd prime a(p^e) = 1 if p == 1 (mod 4) and (p^(e+1) - p^e + 2*(-1)^e)/(p+1) otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*G/Pi^2 = 0.278420154533..., and G is Catalan's constant (A006752). (End)
Conjecture: a(n) = Sum_{k=1..n} sin(GCD(k,n) * Pi/2). - Velin Yanev and Vaclav Kotesovec, Jun 01 2024

A350162 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^2.

Original entry on oeis.org

1, 4, 8, 15, 25, 33, 45, 60, 73, 95, 115, 131, 157, 181, 205, 236, 270, 297, 333, 379, 403, 443, 487, 519, 578, 632, 672, 720, 778, 826, 886, 949, 989, 1059, 1131, 1186, 1260, 1332, 1388, 1482, 1564, 1612, 1696, 1776, 1858, 1946, 2038, 2102, 2187, 2308, 2380, 2490

Offset: 1

Seiichi Manyama, Dec 18 2021

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000

Column 2 of A350161.

Cf. A002654, A014200, A050469, A101455, A344720, A350143, A350166.

a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 18 2021 *)

a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^2);

a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(2*d-1)));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1+x^(2*k)))/(1-x))

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (2*d - 1) = Sum_{k=1..n} 2 * A050469(k) - A002654(k) = 2 * A350166(n) - A014200(n).

G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 + x^(2*k)).

cp's OEIS Frontend

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

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A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

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A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

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Maple

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A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

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A364012 Expansion of Sum_{k>0} k * x^k / (1 + x^(3*k)).

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A322084 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

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A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

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A327122 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 + x^(2*k)), where sigma = A000203.

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