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 1-8 of 8 results.

A144366 Shifts 2 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 4, 12, 8, 28, 17, 60, 34, 134, 68, 276, 140, 580, 280, 1186, 560, 2436, 1128, 4906, 2256, 9976, 4516, 20020, 9048, 40324, 18096, 80860, 36192, 162320, 72418, 324920, 144852, 651177, 289704, 1302914, 579476, 2608360, 1158952
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Comments

Appears to be essentially the same as a sequence in Section 2.2 of Baril-Ramirez. - N. J. A. Sloane, Feb 18 2024

Links

Crossrefs

2nd column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 2: with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq(a(n), n=1..50);
  • Mathematica
    k = 2; dck[b_, c_][n_, k_] := dck[b, c][n, k] = DivisorSum[n, b[#, k] * c[n/#, k]&]; B = dck[T, T]; T[n_, k_] :=  If[n <= k, 1, B[n-k, k]]; a[n_] := T[n, k]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Apr 05 2017, translated from Maple *)

Formula

G.f.: x + x^2 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144367 Shifts 3 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 10, 13, 16, 20, 32, 32, 46, 68, 73, 92, 152, 146, 200, 310, 312, 400, 658, 628, 832, 1328, 1302, 1664, 2740, 2604, 3400, 5500, 5300, 6812, 11178, 10600, 13770, 22388, 21412, 27540, 45132, 42824, 55392, 90352, 86048, 110784
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

3rd column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 3: with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq(a(n), n=1..55);
  • Mathematica
    dck[b_, c_][n_, k_] := dck[b, c][n, k] = Sum[b[d, k]*c[n/d, k], {d, If[n < 0, {}, Divisors[n]]}];
B = dck[T, T];
T[n_, k_] := If[n <= k, 1, B[n - k, k]];
a[n_] := T[n, 3];
Array[a, 55] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

G.f.: x + x^2 + x^3 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144368 Shifts 4 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 2, 6, 4, 8, 5, 14, 8, 22, 10, 32, 18, 51, 20, 72, 36, 116, 44, 152, 72, 258, 89, 314, 148, 548, 178, 660, 296, 1146, 364, 1340, 596, 2380, 728, 2716, 1202, 4880, 1456, 5508, 2404, 9912, 2932, 11088, 4808, 20128, 5868, 22276, 9636
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

4th column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 4: with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq(a(n), n=1..55);
  • Mathematica
    dck[b_, c_][n_, k_] := dck[b, c][n, k] = Sum[b[d, k]*c[n/d, k], {d, If[n < 0, {}, Divisors[n]]}];
B = dck[T, T];
T[n_, k_] := If[n <= k, 1, B[n - k, k]];
a[n_] := T[n, 4];
Array[a, 55] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

G.f.: x + x^2 + x^3 + x^4 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144369 Shifts 5 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 6, 7, 6, 8, 12, 12, 18, 14, 21, 24, 32, 36, 34, 46, 56, 64, 86, 69, 104, 118, 146, 172, 156, 208, 256, 300, 368, 316, 455, 512, 636, 748, 668, 910, 1084, 1272, 1552, 1354, 1884, 2168, 2644, 3108, 2780, 3792, 4440, 5288, 6358, 5568
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

5th column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 5: with (numtheory): dck:= proc(b,c) proc(n, k) option remember; add (b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq (a(n), n=1..60);

Formula

G.f.: x + x^2 + x^3 + x^4 + x^5 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144370 Shifts 6 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 2, 6, 5, 8, 4, 12, 4, 14, 12, 21, 8, 30, 8, 36, 26, 46, 16, 74, 17, 76, 56, 106, 32, 166, 34, 172, 116, 220, 66, 369, 68, 352, 236, 478, 132, 776, 136, 750, 486, 972, 264, 1640, 273, 1522, 980, 2020, 528, 3360, 550, 3152, 1968, 4072
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

6th column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 6: with (numtheory): dck:= proc(b,c) proc(n, k) option remember; add (b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq (a(n), n=1..70);

Formula

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144371 Shifts 7 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 6, 8, 8, 6, 10, 13, 12, 18, 16, 22, 14, 26, 26, 32, 37, 40, 48, 34, 52, 66, 64, 86, 86, 108, 70, 125, 132, 144, 180, 194, 216, 158, 250, 290, 300, 386, 388, 472, 317, 540, 592, 640, 772, 836, 950, 668, 1096, 1236, 1280
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

7th column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 7: with (numtheory): dck:= proc(b,c) proc(n, k) option remember; add (b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq (a(n), n=1..80);

Formula

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144372 Shifts 8 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 4, 10, 4, 10, 6, 11, 6, 16, 8, 26, 10, 24, 12, 32, 13, 36, 18, 62, 20, 58, 24, 74, 30, 78, 38, 145, 40, 124, 52, 174, 60, 174, 76, 314, 86, 260, 104, 386, 121, 364, 158, 664, 172, 550, 212, 834, 250, 748, 316, 1410, 344, 1124
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

8th column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 8: with (numtheory): dck:= proc(b,c) proc(n, k) option remember; add (b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq (a(n), n=1..80);

Formula

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019

A144373 Shifts 9 places left under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 4, 8, 6, 6, 10, 7, 8, 10, 8, 12, 18, 16, 12, 28, 15, 22, 22, 22, 24, 48, 32, 30, 60, 38, 46, 57, 44, 56, 102, 76, 60, 142, 76, 108, 124, 100, 112, 234, 153, 136, 292, 174, 216, 276, 204, 246, 476, 330, 272, 642, 348, 464
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Links

Crossrefs

9th column of A144374. Cf. A000005.

Programs

  • Maple
    k:= 9: with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: a:= n-> T(n,k): seq(a(n), n=1..80);

Formula

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 09 2019
Showing 1-8 of 8 results.

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