A321565 -id:A321565 - cp's OEIS frontend

A321560 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^4.

1, -17, 82, -241, 626, -1394, 2402, -3825, 6643, -10642, 14642, -19762, 28562, -40834, 51332, -61169, 83522, -112931, 130322, -150866, 196964, -248914, 279842, -313650, 391251, -485554, 538084, -578882, 707282, -872644, 923522, -978673, 1200644

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.

Column k=4 of A322083.

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^4*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^4 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321560(n)=sumdiv(n, d, (-1)^(n\d-d)*d^4), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^4*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^4*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -(7*2^(4*e+1) + 31)/15, and a(p^e) = (p^(4*e+4) - 1)/(p^4 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321561 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^5.

Original entry on oeis.org

1, -33, 244, -993, 3126, -8052, 16808, -31713, 59293, -103158, 161052, -242292, 371294, -554664, 762744, -1014753, 1419858, -1956669, 2476100, -3104118, 4101152, -5314716, 6436344, -7737972, 9768751, -12252702, 14408200, -16690344, 20511150

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.

Column k=5 of A322083.

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^5*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^5 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321561(n)=sumdiv(n, d, (-1)^(n\d-d)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^5*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^5*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -3*(5*2^(5*e+1) + 21)/31, and a(p^e) = (p^(5*e+5) - 1)/(p^5 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321562 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^6.

Original entry on oeis.org

1, -65, 730, -4033, 15626, -47450, 117650, -257985, 532171, -1015690, 1771562, -2944090, 4826810, -7647250, 11406980, -16510913, 24137570, -34591115, 47045882, -63019658, 85884500, -115151530, 148035890, -188329050, 244156251, -313742650

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.

Column k=6 of A322083.

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^6*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^6 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321562(n)=sumdiv(n, d, (-1)^(n\d-d)*d^6), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^6*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^6*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -(31*2^(6*e+1) + 127)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321563 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^7.

Original entry on oeis.org

1, -129, 2188, -16257, 78126, -282252, 823544, -2080641, 4785157, -10078254, 19487172, -35570316, 62748518, -106237176, 170939688, -266321793, 410338674, -617285253, 893871740, -1270094382, 1801914272, -2513845188, 3404825448, -4552442508

Offset: 1

N. J. A. Sloane, Nov 23 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.

Column k=7 of A322083.

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^7 &]; Array[a, 25] (* Amiram Eldar, Nov 22 2022 *)

apply( A321563(n)=sumdiv(n, d, (-1)^(n\d-d)*d^7), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^7*x^k/(1 + x^k). - Ilya Gutkovskiy, Dec 22 2018

Multiplicative with a(2^e) = -3*(21*2^(7*e+1) + 85)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321809 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^12.

Original entry on oeis.org

1, -4097, 531442, -16773121, 244140626, -2177317874, 13841287202, -68702695425, 282430067923, -1000244144722, 3138428376722, -8913940970482, 23298085122482, -56707753666594, 129746582562692, -281406240452609, 582622237229762

Offset: 1

N. J. A. Sloane, Nov 23 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.

Column k=12 of A322083.

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^12 &]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

apply( A321809(n)=sumdiv(n, d, (-1)^(n\d-d)*d^12), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^12*x^k/(1 + x^k). - Ilya Gutkovskiy, Dec 22 2018

Multiplicative with a(2^e) = -(2047*2^(12*e+1) + 8191)/4095, and a(p^e) = (p^(12*e+12) - 1)/(p^12 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321815 Sum of 11th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 177148, 1, 48828126, 177148, 1977326744, 1, 31381236757, 48828126, 285311670612, 177148, 1792160394038, 1977326744, 8649804864648, 1, 34271896307634, 31381236757, 116490258898220, 48828126, 350279478046112, 285311670612, 952809757913928

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=11 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013670, A013959.

List(List(List([1..25],j->DivisorsInt(j)),i->Filtered(i,k->IsOddInt(k))),m->Sum(m,n->n^11)); # Muniru A Asiru, Dec 07 2018

a[n_] := DivisorSum[n, #^11&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321815(n)=sigma(n>>valuation(n,2),11), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321815(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),11)) # Chai Wah Wu, Jul 16 2022

a(n) = A013959(A000265(n)) = sigma_11(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)^11*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^(11*e+11)-1)/(p^11-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^12, where c = zeta(12)/24 = 691*Pi^12/15324309000 = 0.0416769... . (End)

A321817 a(n) = Sum_{d|n, n/d odd} d^6 for n > 0.

Original entry on oeis.org

1, 64, 730, 4096, 15626, 46720, 117650, 262144, 532171, 1000064, 1771562, 2990080, 4826810, 7529600, 11406980, 16777216, 24137570, 34058944, 47045882, 64004096, 85884500, 113379968, 148035890, 191365120, 244156251, 308915840, 387952660

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. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A013665.

a[n_] := DivisorSum[n, #^6 &, OddQ[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 02 2022 *)

apply( A321817(n)=sumdiv(n,d,if(bittest(n\d,0),d^6)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^6*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(6*e) 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 = 127*zeta(7)/896 = 0.142924... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-6)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

A321818 a(n) = Sum_{d|n, n/d odd} d^8 for n > 0.

Original entry on oeis.org

1, 256, 6562, 65536, 390626, 1679872, 5764802, 16777216, 43053283, 100000256, 214358882, 430047232, 815730722, 1475789312, 2563287812, 4294967296, 6975757442, 11021640448, 16983563042, 25600065536, 37828630724, 54875873792, 78310985282

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. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A013667.

a[n_] := DivisorSum[n, #^8 &, OddQ[n/#] &]; Array[a, 24] (* Amiram Eldar, Nov 02 2022 *)

apply( A321818(n)=sumdiv(n,d,if(bittest(n\d,0),d^8)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^8*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(8*e) and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = 511*zeta(9)/4608 = 0.1111168... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-8)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A321819 a(n) = Sum_{d|n, n/d odd} d^10 for n > 0.

Original entry on oeis.org

1, 1024, 59050, 1048576, 9765626, 60467200, 282475250, 1073741824, 3486843451, 10000001024, 25937424602, 61918412800, 137858491850, 289254656000, 576660215300, 1099511627776, 2015993900450, 3570527693824, 6131066257802, 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. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A013669.

a[n_] := DivisorSum[n, #^10 &, OddQ[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)

apply( A321819(n)=sumdiv(n,d,if(bittest(n\d,0),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, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(10*e) and a(p^e) = (p^(10*e+10)-1)/(p^10-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^11, where c = 2047*zeta(11)/22528 = 0.090909606... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-10)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A321820 a(n) = Sum_{d|n, n/d odd} d^12 for n > 0.

Original entry on oeis.org

1, 4096, 531442, 16777216, 244140626, 2176786432, 13841287202, 68719476736, 282430067923, 1000000004096, 3138428376722, 8916117225472, 23298085122482, 56693912379392, 129746582562692, 281474976710656, 582622237229762, 1156833558212608

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. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A013671.

a[n_] := DivisorSum[n, #^12 &, OddQ[n/#] &]; Array[a, 20] (* Amiram Eldar, Nov 02 2022 *)

apply( A321820(n)=sumdiv(n,d,if(bittest(n\d,0),d^12)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^12*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(12*e) 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 = 8191*zeta(13)/106496 = 0.0769231... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-12)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

cp's OEIS Frontend

A321560 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^4.

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A321561 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^5.

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A321562 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^6.

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A321563 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^7.

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A321809 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^12.

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A321815 Sum of 11th powers of odd divisors of n.

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A321817 a(n) = Sum_{d|n, n/d odd} d^6 for n > 0.

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A321818 a(n) = Sum_{d|n, n/d odd} d^8 for n > 0.