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 11-15 of 15 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=4 of A322083.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Magma
    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
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^4 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)
  • PARI
    apply( A321560(n)=sumdiv(n, d, (-1)^(n\d-d)*d^4), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Sage
    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

Formula

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

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=5 of A322083.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Magma
    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
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^5 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)
  • PARI
    apply( A321561(n)=sumdiv(n, d, (-1)^(n\d-d)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Sage
    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

Formula

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

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=6 of A322083.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Magma
    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
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^6 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)
  • PARI
    apply( A321562(n)=sumdiv(n, d, (-1)^(n\d-d)*d^6), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Sage
    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

Formula

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

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=7 of A322083.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^7 &]; Array[a, 25] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    apply( A321563(n)=sumdiv(n, d, (-1)^(n\d-d)*d^7), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

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

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=12 of A322083.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^12 &]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    apply( A321809(n)=sumdiv(n, d, (-1)^(n\d-d)*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 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
Previous Showing 11-15 of 15 results.

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

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