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.

Showing 1-6 of 6 results.

A128378 Partial sums of A128379.

Original entry on oeis.org

1, 0, -24, -24, 252, 552, -1196, -5496, -1218, 27808, 50532, -44136, -286534, -305256, 551724, 2023976, 1639485, -3659784, -11484752, -9396720, 16258722, 55073200, 55004040, -44731872, -220443155, -289179552, 5590128, 691963304, 1254552228, 741227832, -1414045956
Offset: 1

Views

Author

Gary W. Adamson, Feb 28 2007

Keywords

Comments

Conjecture: Given the infinite set of sequences generated from using the partial sum operator on A000594, (i.e. A000012^k * A000594, k in succession = 1, 2, 3, ...), k = 23 and k = 24 are the only two sequences in the set with zeros. In A128379, k = 23: (1, -1, -24, 0, 276, 300, -1748, -4300, ...).

Crossrefs

Cf. A000012, A000594, A128379.

Programs

  • Mathematica
    Nest[Accumulate, RamanujanTau[Range[32]], 24] (* Amiram Eldar, Jan 08 2025 *)

Formula

A000012^24 * A000594; (partial sum operator performed 24 times on A000594).

A128380 A097806^24 * A000594.

Original entry on oeis.org

1, 0, -48, -24, 1104, 1128, -15892, -25368, 156240, 360640, -1057908, -3600696, 4417678, 26438568, -3155508, -144207816, -112109568, 564538680, 1002957320, -1344487080, -5096138658, -111333800, 17182425012, 17552839368, -34668142443, -86942440944, 4993723500, 236551774320
Offset: 1

Views

Author

Gary W. Adamson, Feb 28 2007

Keywords

Comments

Conjecture: Given the infinite set of sequences generated from the pairwise operation on A000594 (A097806^k * A000594), k = 24, (A128380) is the only sequence in the set with a zero. The sequence generated from k=23 = (1, -1, -47, 23, 1081, 47, -15939, ...). Analogous conjecture with the partial sum operator: (Cf. A128378, A128379); in which zeros are conjectured to occur only with k=23 and k=24. A128380 mod 24 = 1, 0, 0, 0, 0, 0, -4, 0, 0, 16, ...

Crossrefs

Cf. A000594, A097806, A128378, A128379.

Programs

  • Mathematica
    Nest[Prepend[Most[#] + Rest[#], First[#]] &, RamanujanTau[Range[30]], 24] (* Amiram Eldar, Jan 08 2025 *)

Formula

Pairwise operation performed 24 times on A000594

Extensions

More terms from Amiram Eldar, Jan 08 2025

A128381 A007318^24 * A000594.

Original entry on oeis.org

1, 0, -324, -10976, -260898, -4919184, -67536616, -212659776, 28757879829, 1419855850728, 48139832128404, 1387340166285216, 36039677403855158, 861269016060967824, 18976682736434056968, 379832429643337648960, 6586595998856413482930, 83878202724395340243384, 7117837083803882256428
Offset: 0

Views

Author

Gary W. Adamson, Feb 28 2007

Keywords

Comments

Conjecture: given the bto performed any k times on A000594 (k = 1, 2, 3, ...); k = 6 and k = 24 are the only members of the set with zeros. k = 6 generates (1, -18, 0, 688, 4494, 5508, ...).

Crossrefs

Cf. A007318, A000594, A128378, A128379, A128380, A128382.

Programs

  • Mathematica
    Nest[Table[Sum[Binomial[n-1, k-1] * #[[k]], {k, 1, n}], {n, 1, Length[#]}] &, RamanujanTau[Range[19]], 24] (* Amiram Eldar, Jan 08 2025 *)

Formula

Binomial transform operation performed 24 times on A000594, assuming A000594 has offset zero.

A128382 Inverse Moebius transform operation performed 24 times on A000594: A051731^24 * A000594.

Original entry on oeis.org

1, 0, 276, -1748, 4854, 0, -16720, 44552, -107295, 0, 534636, -482448, -577714, 0, 1339704, 2528206, -6905910, 0, 10661444, -8484792, -4614720, 0, 18643296, 12296352, -25383005, 0, -75928312, 29226560, 128406654, 0, -52843144, -151821160, 147559536, 0, -81158880
Offset: 1

Views

Author

Gary W. Adamson, Feb 28 2007

Keywords

Comments

Conjecture: given the inverse Moebius transform operation performed any k times (k=1,2,3,...); k=24 is the only such sequence with zeros. A weaker conjecture: "zero" occurs an infinite number of times in A128382.
Multiplicative because A000594 is. Each application of A051731 corresponds to an inverse Moebius transform. - Andrew Howroyd, Aug 03 2018

Links

Crossrefs

Cf. A051731, A000594, A128378, A128379, A128380, A128381.

Programs

  • Mathematica
    nmax = 40;
M = Table[If[Mod[n, k] == 0, 1, 0], {n, nmax}, {k, nmax}];
MatrixPower[M, 24].RamanujanTau[Range[nmax]] (* Jean-François Alcover, Sep 20 2019 *)
  • PARI
    seq(n, k=24)={my(u=vector(n,n,1), v=vector(n,n,ramanujantau(n))); for(i=1, k, v=dirmul(u,v)); v} \\ Andrew Howroyd, Aug 03 2018

Formula

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = (Sum_{n>=1} A000594(n)/n^s)*zeta(s)^24. - Jianing Song, Aug 04 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A128391 A054523^24 * A000594.

Original entry on oeis.org

1, 0, 300, -1724, 4926, 0, -16600, 44600, -100299, 0, 534852, -517200, -577450, 0, 1477800, 2486626, -6905550, 0, 10661852, -8492424, -4980000, 0, 18643800, 13380000, -25030649, 0, -78396200, 28618400, 128407302, 0, -52842448, -150834520, 160455600, 0, -81771600
Offset: 1

Views

Author

Gary W. Adamson, Feb 28 2007

Keywords

Comments

Conjecture: Given A054523^k, k = any positive integer, "zero" appears only in the sequence A018391 (k=24).
Each application of A054523 corresponds to the Dirichlet convolution of A000010 with the sequence on the right. Since both A000594 and A000010 are multiplicative, the resulting sequence will also be multiplicative. - Andrew Howroyd, Aug 03 2018

Links

Crossrefs

Cf. A000010, A000594, A054523, A128378, A128379, A128380, A128381, A128392.

Programs

  • Mathematica
    nmax = 40;
T[n_, k_] := If[Divisible[n, k], EulerPhi[n/k], 0]; T[1, 1] = 1;
M = Table[T[n, k], {n, 1, nmax}, {k, 1, nmax}];
MatrixPower[M, 24].RamanujanTau[Range[nmax]] (* Jean-François Alcover, Sep 20 2019 *)
  • PARI
    seq(n, k=24)={my(u=vector(n,n,eulerphi(n)), v=vector(n,n,ramanujantau(n))); for(i=1, k, v=dirmul(u,v)); v} \\ Andrew Howroyd, Aug 03 2018

Formula

A054523^24 as an infinite lower triangular matrix * A000594.

Extensions

a(7) corrected and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A128392 A126988^12 * A000594.

Original entry on oeis.org

1, 0, 288, -1736, 4890, 0, -16660, 44576, -103869, 0, 534744, -499968, -577582, 0, 1408320, 2507344, -6905730, 0, 10661648, -8489040, -4798080, 0, 18643548, 12837888, -25207475, 0, -77183496, 28921760, 128406978, 0, -52842796, -151328128, 154006272, 0, -81467400
Offset: 1

Views

Author

Gary W. Adamson, Feb 28 2007

Keywords

Comments

Conjecture: Given A126988^k, k any positive integer, A128392 is the only sequence in the infinite set with zeros.
Each application of A126988 corresponds to the Dirichlet convolution of the natural numbers with the sequence on the right. Since both Ramanujan's tau function A000594 and the natural numbers are multiplicative, the resulting sequence will also be multiplicative. - Andrew Howroyd, Aug 03 2018

Links

Crossrefs

Cf. A126988, A000594, A128378, A128379, A128380, A128381, A128391.

Programs

  • Mathematica
    nmax = 40;
M = Table[If[Mod[n, m] == 0, n/m, 0], {n, 1, nmax}, {m, 1, nmax}];
MatrixPower[M, 12].RamanujanTau[Range[nmax]] (* Jean-François Alcover, Sep 20 2019 *)
  • PARI
    seq(n, k=12)={my(u=vector(n,n,n), v=vector(n,n,ramanujantau(n))); for(i=1, k, v=dirmul(u,v)); v} \\ Andrew Howroyd, Aug 03 2018

Formula

A126988 as an infinite lower triangular matrix, * A000594.

Extensions

Terms a(11) and beyond from Andrew Howroyd, Aug 03 2018
Showing 1-6 of 6 results.

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