A050471 -id:A050471 - cp's OEIS frontend

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

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

A050462 a(n) = Sum_{d|n, n/d=1 mod 4} d^3.

Original entry on oeis.org

1, 8, 27, 64, 126, 216, 343, 512, 730, 1008, 1331, 1728, 2198, 2744, 3402, 4096, 4914, 5840, 6859, 8064, 9262, 10648, 12167, 13824, 15751, 17584, 19710, 21952, 24390, 27216, 29791, 32768, 35938, 39312, 43218, 46720, 50654, 54872, 59346

Offset: 1

N. J. A. Sloane, Dec 23 1999

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A007331, A050466, A050471, A175572.

Cf. A050460, A050461, A050463.

a[n_] := Total[(n/Select[Divisors@ n, Mod[#, 4] == 1 &])^3]; Array[a, 39] (* Robert G. Wilson v, Mar 26 2015 *)
a[n_] := DivisorSum[n, #^3 &, Mod[n/#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

a(n) = sumdiv(n, d, ((n/d % 4)== 1)* d^3); \\ Michel Marcus, Mar 26 2015

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A007331(n) - A050466(n).
a(n) = A050471(n) + A050466(n).
a(n) = (A007331(n) + A050471(n))/2.
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = Pi^4/192 + A175572/2 = 1.00181129167264... . (End)

Offset changed from 0 to 1 by Robert G. Wilson v, Mar 27 2015

A050466 a(n) = Sum_{d|n, n/d=3 mod 4} d^3.

Original entry on oeis.org

0, 0, 1, 0, 0, 8, 1, 0, 27, 0, 1, 64, 0, 8, 126, 0, 0, 216, 1, 0, 370, 8, 1, 512, 0, 0, 730, 64, 0, 1008, 1, 0, 1358, 0, 126, 1728, 0, 8, 2198, 0, 0, 2960, 1, 64, 3402, 8, 1, 4096, 343, 0, 4914, 0, 0, 5840, 126, 512, 6886, 0, 1, 8064, 0, 8, 9991, 0

Offset: 1

N. J. A. Sloane, Dec 23 1999

From Robert G. Wilson v, Mar 26 2015: (Start)
a(n) = 0 for n = 1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, ... (A072437).
a(n) = 1 for n = 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ... (A002145). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A007331, A050462, A050471, A175572.
Cf. A050464, A050465, A050467.
Cf. A002145, A072437.

a[n_] := Total[(n/Select[Divisors@ n, Mod[#, 4] == 3 &])^3]; Array[a, 64] (* Robert G. Wilson v, Mar 26 2015 *)
a[n_] := DivisorSum[n, #^3 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

a(n) = sumdiv(n, d, ((n/d % 4)== 3)* d^3); \\ Michel Marcus, Mar 26 2015

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A007331(n) - A050462(n).
a(n) = A050462(n) - A050471(n).
a(n) = (A007331(n) - A050471(n))/2.
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = Pi^4/192 - A175572/2 = 0.0128667399315... . (End)

Offset changed from 0 to 1 by Robert G. Wilson v, Mar 27 2015

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)).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A050462 a(n) = Sum_{d|n, n/d=1 mod 4} d^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A050466 a(n) = Sum_{d|n, n/d=3 mod 4} d^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula