cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 51 results. Next

A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

Original entry on oeis.org

1, 127, 2188, 16255, 78126, 277876, 823544, 2080639, 4785157, 9922002, 19487172, 35565940, 62748518, 104590088, 170939688, 266321791, 410338674, 607714939, 893871740, 1269938130, 1801914272, 2474870844, 3404825448, 4552438132, 6103593751, 7969061786, 10465138360, 13386707720, 17249876310
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Sum_{k>=1} k^b*x^k/(1 + x^k): A000593 (b=1), A078306 (b=2), A078307 (b=3), A284900 (b=4), A284926 (b=5), A284927 (b=6), this sequence (b=7), A321553 (b=8), A321554 (b=9), A321555 (b=10), A321556 (b=11), A321557 (b=12).
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013666.

Programs

  • Mathematica
    f[p_, e_] := (p^(7*e + 7) - 1)/(p^7 - 1); f[2, e_] := (63*2^(7*e + 1) + 1)/127; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
  • PARI
    apply( A321552(n)=sumdiv(n, d, (-1)^(n\d-1)*d^7), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (63*2^(7*e+1)+1)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 127*zeta(8)/1024 = 0.124529... . (End)

A321557 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^12.

Original entry on oeis.org

1, 4095, 531442, 16773119, 244140626, 2176254990, 13841287202, 68702695423, 282430067923, 999755863470, 3138428376722, 8913939907598, 23298085122482, 56680071092190, 129746582562692, 281406240452607, 582622237229762, 1156551128144685, 2213314919066162, 4094999772632494
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013671.

Programs

  • Mathematica
    f[p_, e_] := (p^(12*e + 12) - 1)/(p^12 - 1); f[2, e_] := (2047*2^(12*e + 1) + 1)/4095; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
  • PARI
    apply( A321557(n)=sumdiv(n, d, (-1)^(n\d-1)*d^12), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^12*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (2047*2^(12*e+12)+1)/4095, and a(p^e) = (p^(12*e+12) - 1)/(p^12 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^13, where c = 315*zeta(13)/4096 = 0.0769137... . (End)

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

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^12&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)
  • PARI
    apply( A321816(n)=sigma(n>>valuation(n,2),12), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Python
    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

Formula

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)

A321553 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^8.

Original entry on oeis.org

1, 255, 6562, 65279, 390626, 1673310, 5764802, 16711423, 43053283, 99609630, 214358882, 428360798, 815730722, 1470024510, 2563287812, 4278124287, 6975757442, 10978587165, 16983563042, 25499674654, 37828630724, 54661514910, 78310985282, 109660357726, 152588281251, 208011334110, 282472589764
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013667.

Programs

  • Mathematica
    Table[Total[(-1)^(n/#+1) #^8&/@Divisors[n]],{n,30}] (* Harvey P. Dale, May 05 2021 *)
f[p_, e_] := (p^(8*e + 8) - 1)/(p^8 - 1); f[2, e_] := (127*2^(8*e + 1) + 1)/255; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
  • PARI
    apply( A321553(n)=sumdiv(n, d, (-1)^(n\d-1)*d^8), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^8*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (127*2^(8*e+1)+1)/255, and a(p^e) = (p^(8*e+8) - 1)/(p^8 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = 85*zeta(9)/768 = 0.110899... . (End)

A321554 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^9.

Original entry on oeis.org

1, 511, 19684, 261631, 1953126, 10058524, 40353608, 133955071, 387440173, 998047386, 2357947692, 5149944604, 10604499374, 20620693688, 38445332184, 68584996351, 118587876498, 197981928403, 322687697780, 510998308506, 794320419872, 1204911270612, 1801152661464, 2636771617564, 3814699218751
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013668.

Programs

  • Mathematica
    f[p_, e_] := (p^(9*e + 9) - 1)/(p^9 - 1); f[2, e_] := (255*2^(9*e + 1) + 1)/511; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
  • PARI
    apply( A321554(n)=sumdiv(n, d, (-1)^(n\d-1)*d^9), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^9*x^k/(1 + x^k). - Seiichi Manyama, Nov 24 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (255*2^(9*e+1)+1)/511, and a(p^e) = (p^(9*e+9) - 1)/(p^9 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = 511*zeta(10)/5120 = 0.0999039... . (End)

A321555 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^10.

Original entry on oeis.org

1, 1023, 59050, 1047551, 9765626, 60408150, 282475250, 1072692223, 3486843451, 9990235398, 25937424602, 61857886550, 137858491850, 288972180750, 576660215300, 1098436836351, 2015993900450, 3567040850373, 6131066257802, 10229991281926, 16680163512500, 26533985367846, 41426511213650, 63342475768150
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013669.

Programs

  • Mathematica
    f[p_, e_] := (p^(10*e + 10) - 1)/(p^10 - 1); f[2, e_] := (511*2^(10*e + 1) + 1)/1023; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
  • PARI
    apply( A321555(n)=sumdiv(n, d, (-1)^(n\d-1)*d^10), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (511*2^(10*e+1)+1)/1023, and a(p^e) = (p^(10*e+10) - 1)/(p^10 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^11, where c = 93*zeta(11)/1024 = 0.0908651... . (End)

A321556 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^11.

Original entry on oeis.org

1, 2047, 177148, 4192255, 48828126, 362621956, 1977326744, 8585738239, 31381236757, 99951173922, 285311670612, 742649588740, 1792160394038, 4047587844968, 8649804864648, 17583591913471, 34271896307634, 64237391641579
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013670.

Programs

  • Mathematica
    f[p_, e_] := (p^(11*e + 11) - 1)/(p^11 - 1); f[2, e_] := (1023*2^(11*e + 1) + 1)/2047; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)
  • PARI
    apply( A321556(n)=sumdiv(n, d, (-1)^(n\d-1)*d^11), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (1023*2^(11*e+1)+1)/2047, and a(p^e) = (p^(11*e+11) - 1)/(p^11 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^12, where c = 2047*zeta(12)/24576 = 0.0833131... . (End)

A321558 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^2.

Original entry on oeis.org

1, -5, 10, -13, 26, -50, 50, -45, 91, -130, 122, -130, 170, -250, 260, -173, 290, -455, 362, -338, 500, -610, 530, -450, 651, -850, 820, -650, 842, -1300, 962, -685, 1220, -1450, 1300, -1183, 1370, -1810, 1700, -1170, 1682, -2500, 1850, -1586, 2366
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Examples

			G.f. = x - 5*x^2 + 10*x^3 - 13*x^4 + 26*x^5 - 50*x^6 + 50*x^7 + ... - _Michael Somos_, Oct 24 2019

Links

Crossrefs

Column k=2 of A322083.
Glaisher's xi_i (i=0..12): A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809
Cf. A321543 - A321557, A321810 - A321836 for similar sequences.
Cf. A078306, A328667.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^2*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^2 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)
  • PARI
    apply( A321558(n)=sumdiv(n, d, (-1)^(n\d-d)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Sage
    s=(sum((-1)^(k+1)*k^2*x^k/(1 + x^k)  for k in (1..50))).series(x, 30); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*k^2*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018
G.f.: Sum_{k>=1} (-1)^(k+1)*(x^k - x^(2*k))/(1 + x^k)^3. - Michael Somos, Oct 24 2019
a(n) = -(-1)^n A328667(n). a(2*n + 1) = A078306(2*n + 1). a(2*n) = A078306(2*n) - 8*A078306(n). - Michael Somos, Oct 24 2019
From Peter Bala, Jan 29 2022: (Start)
Multiplicative with a(2^k) = - (2^(2*k+1) + 7)/3 for k >= 1 and a(p^k) = (p^(2*k+2) - 1)/(p^2 - 1) for odd prime p.
n^2 = (-1)^(n+1)*Sum_{d divides n} A067856(n/d)*a(d). (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

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^7 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)
  • PARI
    apply( A321811(n)=sigma(n>>valuation(n,2),7), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Python
    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

Formula

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)

A321551 a(n) = Sum_{d|n} (-1)^(d-1)*d^12.

Original entry on oeis.org

1, -4095, 531442, -16781311, 244140626, -2176254990, 13841287202, -68736258047, 282430067923, -999755863470, 3138428376722, -8918293480462, 23298085122482, -56680071092190, 129746582562692, -281543712968703, 582622237229762, -1156551128144685, 2213314919066162, -4096999772640686
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    f[p_, e_] := (p^(12*e + 12) - 1)/(p^12 - 1); f[2, e_] := 2 - (2^(12*e + 12) - 1)/4095; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^12), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^12*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 24 2018
Multiplicative with a(2^e) = 2 - (2^(12*e + 12) - 1)/4095, and a(p^e) = (p^(12*e + 12) - 1)/(p^12 - 1) for p > 2. - Amiram Eldar, Nov 04 2022
Previous Showing 11-20 of 51 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.