A285425 -id:A285425 - cp's OEIS frontend

A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.

1, 1, -2, 1, -3, 2, 1, -5, 4, -1, 1, -9, 10, -3, 2, 1, -17, 28, -13, 6, -4, 1, -33, 82, -57, 26, -12, 2, 1, -65, 244, -241, 126, -50, 8, 0, 1, -129, 730, -993, 626, -252, 50, -3, 3, 1, -257, 2188, -4033, 3126, -1394, 344, -45, 13, -4, 1, -513, 6562, -16257, 15626, -8052, 2402, -441, 91, -18, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

For each k, the k-th column sequence (T(n,k))(n>=1) is a multiplicative function of n, equal to (-1)^(n+1)*(Id_k * 1) in the notation of the Bala link. - Peter Bala, Mar 19 2022

			Square array begins:
   1,   1,   1,    1,     1,     1,  ...
  -2,  -3,  -5,   -9,   -17,   -33,  ...
   2,   4,  10,   28,    82,   244,  ...
  -1,  -3, -13,  -57,  -241,  -993,  ...
   2,   6,  26,  126,   626,  3126,  ...
  -4, -12, -50, -252, -1394, -8052,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Peter Bala, A signed Dirichlet product of arithmetical functions
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322084.

Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
f[p_, e_, k_] := If[k == 0, e + 1, (p^(k*e + k) - 1)/(p^k - 1)]; f[2, e_, k_] := If[k == 0, e - 3, -((2^(k - 1) - 1)*2^(k*e + 1) + 2^(k + 1) - 1)/(2^k - 1)]; T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k, k], {n, 1, 11}, {k, n - 1, 0, -1}] // Flatten (* Amiram Eldar, Nov 22 2022 *)

T(n,k)={sumdiv(n, d, (-1)^(n/d+d)*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} (-1)^(j+1)*j^k*x^j/(1 + x^j).

A321810 Sum of 6th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 730, 1, 15626, 730, 117650, 1, 532171, 15626, 1771562, 730, 4826810, 117650, 11406980, 1, 24137570, 532171, 47045882, 15626, 85884500, 1771562, 148035890, 730, 244156251, 4826810, 387952660, 117650, 594823322, 11406980, 887503682

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).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=6 of A285425.
Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013665, A013954.

f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2022 *)

apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # Chai Wah Wu, Jul 16 2022

a(n) = A013954(A000265(n)) = sigma_6(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)
a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - R. J. Mathar, Aug 15 2023

A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 5, 10, 9, 1, 4, 6, 17, 28, 17, 1, 2, 12, 26, 65, 82, 33, 1, 2, 8, 50, 126, 257, 244, 65, 1, 2, 9, 50, 252, 626, 1025, 730, 129, 1, 4, 10, 65, 344, 1394, 3126, 4097, 2188, 257, 1, 2, 18, 82, 513, 2402, 8052, 15626, 16385, 6562, 513, 1, 4, 12, 130, 730, 4097, 16808, 47450, 78126, 65537, 19684, 1025, 1

Offset: 0

Ilya Gutkovskiy, Aug 02 2017

For row r > 0, Sum_{k=1..n} A(r,k) ~ zeta(r+1) * n^(r+1) / ((r+1) * zeta(r+2)). - Vaclav Kotesovec, May 20 2021

			Square array begins:
1,   2,    2,     2,     2,     4,  ...
1,   3,    4,     5,     6,    12,  ...
1,   5,   10,    17,    26,    50,  ...
1,   9,   28,    65,   126,   252,  ...
1,  17,   82,   257,   626,  1394,  ...
1,  33,  244,  1025,  3126,  8052,  ...

Eric Weisstein's World of Mathematics, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Index entries for sequences related to sums of divisors

Rows n=0-8 give: A034444, A034448, A034676, A034677, A034678, A034679, A034680, A034681, A034682.

Cf. A077610, A109974, A285425.

Dirichlet g.f. of row n: zeta(s)*zeta(s-n)/zeta(2*s-n).

A321816 Sum of 12th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 531442, 1, 244140626, 531442, 13841287202, 1, 282430067923, 244140626, 3138428376722, 531442, 23298085122482, 13841287202, 129746582562692, 1, 582622237229762, 282430067923, 2213314919066162, 244140626, 7355841353205284, 3138428376722

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).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=12 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321815 (analog for 2nd .. 11th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013671, A013960.

a[n_] := DivisorSum[n, #^12&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321816(n)=sigma(n>>valuation(n,2),12), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321816(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),12)) # Chai Wah Wu, Jul 16 2022

a(n) = A013960(A000265(n)) = sigma_12(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^12*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(12*e+12)-1)/(p^12-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^13, where c = zeta(13)/26 = 0.0384662... . (End)

A321811 Sum of 7th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 2188, 1, 78126, 2188, 823544, 1, 4785157, 78126, 19487172, 2188, 62748518, 823544, 170939688, 1, 410338674, 4785157, 893871740, 78126, 1801914272, 19487172, 3404825448, 2188, 6103593751, 62748518, 10465138360, 823544, 17249876310

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).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=7 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013666, A013955.

a[n_] := DivisorSum[n, #^7 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321811(n)=sigma(n>>valuation(n,2),7), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321811(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),7)) # Chai Wah Wu, Jul 16 2022

a(n) = A013955(A000265(n)) = sigma_7(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^7*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(8)/16 = Pi^8/151200 = 0.0627548... . (End)

A292919 Sum of n-th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 28, 1, 3126, 730, 823544, 1, 387440173, 9765626, 285311670612, 531442, 302875106592254, 678223072850, 437893920912786408, 1, 827240261886336764178, 150094635684419611, 1978419655660313589123980, 95367431640626, 5842587018944528395924761632, 81402749386839761113322

Offset: 1

Ilya Gutkovskiy, Sep 26 2017

Seiichi Manyama, Table of n, a(n) for n = 1..386
Index entries for sequences related to sums of divisors

Diagonal of A285425.

Cf. A000593, A023887, A082245.

f:= proc(n) local t,d;
  t:= n/2^padic:-ordp(n,2);
  add(d^n, d = numtheory:-divisors(t));
end proc:
map(f, [$1..30]); # Robert Israel, Sep 27 2017

Rest[Table[SeriesCoefficient[Sum[(2 k - 1)^n x^(2 k - 1)/(1 - x^(2 k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 22}]]
f[n_] := Plus @@ (Select[Divisors[n], OddQ]^n); Array[f, 22] (* Robert G. Wilson v, Sep 26 2017 *)

a(n) = sumdiv(n, d, if (d%2, d^n)); \\ Michel Marcus, Sep 08 2018

a(n) = [x^n] Sum_{k>=1} (2*k - 1)^n*x^(2*k-1)/(1 - x^(2*k-1)).

a(2^k) = 1.

A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 0, 1, 1, 5, 1, 3, 0, 1, 1, 9, 1, 6, 1, 0, 1, 1, 17, 1, 14, 1, 3, 0, 1, 1, 33, 1, 36, 1, 7, 2, 0, 1, 1, 65, 1, 98, 1, 21, 4, 3, 0, 1, 1, 129, 1, 276, 1, 73, 10, 8, 1, 0, 1, 1, 257, 1, 794, 1, 273, 28, 30, 1, 5

Offset: 1

Ilya Gutkovskiy, Nov 01 2018

A(n,k) is the sum of k-th powers of proper divisors of n.

			Square array begins:
  0,  0,   0,   0,   0,    0,  ...
  1,  1,   1,   1,   1,    1,  ...
  1,  1,   1,   1,   1,    1,  ...
  2,  3,   5,   9,  17,   33,  ...
  1,  1,   1,   1,   1,    1,  ...
  3,  6,  14,  36,  98,  276,  ...

Eric Weisstein's World of Mathematics, Proper divisors
Index entries for sequences related to sums of divisors

Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.

Cf. A109974, A285425, A286880, A321259 (diagonal).

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

G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).
Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).
A(n,k) = 1 if n is prime.

A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, -1, 1, 7, 10, 1, 2, 1, 15, 28, 11, 6, 0, 1, 31, 82, 55, 26, 4, 2, 1, 63, 244, 239, 126, 30, 8, -2, 1, 127, 730, 991, 626, 196, 50, 1, 3, 1, 255, 2188, 4031, 3126, 1230, 344, 43, 13, 0, 1, 511, 6562, 16255, 15626, 7564, 2402, 439, 91, 6, 2, 1, 1023, 19684, 65279, 78126, 45990, 16808, 3823, 757, 78, 12, -2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
   0,  1,   3,    7,    15,    31,  ...
   2,  4,  10,   28,    82,   244,  ...
  -1,  1,  11,   55,   239,   991,  ...
   2,  6,  26,  126,   626,  3126,  ...
   0,  4,  30,  196,  1230,  7564,  ...

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 A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557.

Cf. A109974, A279394, A279396, A285425, A321438 (diagonal), A322082, A322083, A322084.

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

T(n,k)={sumdiv(n, d, (-1)^(n/d+1)*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^j).

A322082 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d odd} d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 10, 4, 2, 1, 16, 28, 16, 6, 2, 1, 32, 82, 64, 26, 8, 2, 1, 64, 244, 256, 126, 40, 8, 1, 1, 128, 730, 1024, 626, 224, 50, 8, 3, 1, 256, 2188, 4096, 3126, 1312, 344, 64, 13, 2, 1, 512, 6562, 16384, 15626, 7808, 2402, 512, 91, 12, 2, 1, 1024, 19684, 65536, 78126, 46720, 16808, 4096, 757, 104, 12, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  1,  2,   4,    8,    16,    32,  ...
  2,  4,  10,   28,    82,   244,  ...
  1,  4,  16,   64,   256,  1024,  ...
  2,  6,  26,  126,   626,  3126,  ...
  2,  8,  40,  224,  1312,  7808,  ...

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 A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820.

Cf. A109974, A279394, A279396, A285425, A322081, A322083, A322084.

Table[Function[k, Sum[Boole[OddQ[n/d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
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(n/d%2, 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)).

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

A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.

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A321810 Sum of 6th powers of odd divisors of n.

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A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

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A321816 Sum of 12th powers of odd divisors of n.

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A321811 Sum of 7th powers of odd divisors of n.

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A292919 Sum of n-th powers of odd divisors of n.

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A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.

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A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

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