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.

A372673 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = phi(k*n) / phi(k).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 3, 4, 4, 1, 1, 2, 2, 4, 2, 1, 2, 2, 4, 4, 4, 6, 1, 1, 3, 2, 4, 3, 6, 4, 1, 2, 2, 4, 5, 4, 6, 8, 6, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 1, 2, 2, 4, 4, 4, 7, 8, 6, 8, 10, 8, 12, 1, 1, 3, 2, 5, 3, 6, 4, 9, 5, 10, 6, 12, 6
Offset: 1

Views

Author

Seiichi Manyama, May 10 2024

Keywords

Examples

			Square array T(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...
  2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, ...
  2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, ...
  4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, ...
  2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 6, ...
  6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, ...
  4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, ...
  6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, ...

Links

Crossrefs

Columns k=1..18 give: A000010, A062570, A195459, A062570, A359101, A372671, A359102, A062570, A195459, A372672, A372676, A372671, A372677, A372678, A372679, A062570, A372681, A372671.
Main diagonal gives A000027.
Cf. A005117, A372619.

Programs

  • PARI
    T(n, k) = eulerphi(k*n)/eulerphi(k);

A372606 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 2, 4, 5, 6, 4, 6, 10, 9, 10, 2, 8, 10, 14, 13, 12, 6, 6, 16, 18, 22, 17, 18, 4, 12, 12, 24, 26, 28, 23, 22, 6, 12, 24, 20, 44, 34, 40, 31, 28, 4, 12, 20, 36, 28, 52, 46, 48, 37, 32, 10, 12, 30, 36, 60, 40, 76, 62, 66, 45, 42, 4, 20, 20, 42, 52, 72, 52, 92, 74, 74, 55, 46
Offset: 1

Views

Author

Seiichi Manyama, May 07 2024

Keywords

Examples

			Square array T(n,k) begins:
   1,  1,  2,  2,  4,  2,   6, ...
   2,  3,  4,  6,  8,  6,  12, ...
   4,  5, 10, 10, 16, 12,  24, ...
   6,  9, 14, 18, 24, 20,  36, ...
  10, 13, 22, 26, 44, 28,  60, ...
  12, 17, 28, 34, 52, 40,  72, ...
  18, 23, 40, 46, 76, 52, 114, ...

Links

Crossrefs

Columns k=1..2 give: A002088, A049690.
Main diagonal gives A372608.
Cf. A000010, A372619.
Cf. A007947, A048250, A332880, A332881.

Programs

  • Mathematica
    T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 10 2024 *)
  • PARI
    T(n, k) = sum(j=1, n, eulerphi(k*j));

Formula

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = k * A007947(k)/A048250(k) = k * A332881(k) / A332880(k) is the multiplicative function defined by c(p^e) = p^(e+1)/(p+1). - Amiram Eldar, May 10 2024

A372636 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j*k) / phi(k).

Original entry on oeis.org

1, 5, 14, 31, 58, 93, 148, 219, 306, 407, 550, 695, 898, 1103, 1323, 1610, 1963, 2293, 2738, 3152, 3597, 4116, 4773, 5362, 6073, 6808, 7611, 8437, 9492, 10348, 11557, 12728, 13868, 15143, 16425, 17753, 19482, 21083, 22687, 24350, 26481, 28186, 30535, 32641
Offset: 1

Views

Author

Seiichi Manyama, May 08 2024

Keywords

Links

Crossrefs

Cf. A330596, A372619, A372633.

Programs

  • Mathematica
    Table[Sum[Sum[EulerPhi[j*k], {j, 1, n}] / EulerPhi[k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 08 2024 *)
s = 1; Join[{1}, Table[s += Sum[EulerPhi[j*n] / EulerPhi[j], {j, 1, n}] + Sum[EulerPhi[j*n], {j, 1, n-1}] / EulerPhi[n], {n, 2, 50}]] (* Vaclav Kotesovec, May 08 2024 *)
  • PARI
    a(n) = sum(j=1, n, sum(k=1, n, eulerphi(j*k)/eulerphi(k)));

Formula

a(n) ~ c * n^3, where c = A330596 / 2 = 0.374267629841... . - Amiram Eldar, May 09 2024

A372621 a(n) = (1/2) * Sum_{k=1..n} phi(3*k).

Original entry on oeis.org

1, 2, 5, 7, 11, 14, 20, 24, 33, 37, 47, 53, 65, 71, 83, 91, 107, 116, 134, 142, 160, 170, 192, 204, 224, 236, 263, 275, 303, 315, 345, 361, 391, 407, 431, 449, 485, 503, 539, 555, 595, 613, 655, 675, 711, 733, 779, 803, 845, 865, 913, 937, 989, 1016, 1056, 1080, 1134
Offset: 1

Views

Author

Seiichi Manyama, May 07 2024

Keywords

Links

Crossrefs

Equals A194881 - 1.
Partial sums of A195459.
Column k=3 of A372619.
Cf. A000010, A194881.

Programs

  • Mathematica
    Accumulate[Table[EulerPhi[3*n], {n, 1, 60}]]/2 (* Amiram Eldar, May 08 2024 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(3*k))/2;

Formula

a(n) ~ (27/(8*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

A372637 a(n) = (1/2) * Sum_{k=1..n} phi(6*k).

Original entry on oeis.org

1, 3, 6, 10, 14, 20, 26, 34, 43, 51, 61, 73, 85, 97, 109, 125, 141, 159, 177, 193, 211, 231, 253, 277, 297, 321, 348, 372, 400, 424, 454, 486, 516, 548, 572, 608, 644, 680, 716, 748, 788, 824, 866, 906, 942, 986, 1032, 1080, 1122, 1162, 1210, 1258, 1310, 1364, 1404
Offset: 1

Views

Author

Seiichi Manyama, May 08 2024

Keywords

Links

Crossrefs

Column k=6 of A372619.
Cf. A000010.

Programs

  • Mathematica
    Accumulate[Table[EulerPhi[6*n], {n, 1, 60}]]/2 (* Amiram Eldar, May 08 2024 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(6*k))/2;

Formula

a(n) ~ (9/(2*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

A372639 a(n) = (1/4) * Sum_{k=1..n} phi(10*k).

Original entry on oeis.org

1, 3, 5, 9, 14, 18, 24, 32, 38, 48, 58, 66, 78, 90, 100, 116, 132, 144, 162, 182, 194, 214, 236, 252, 277, 301, 319, 343, 371, 391, 421, 453, 473, 505, 535, 559, 595, 631, 655, 695, 735, 759, 801, 841, 871, 915, 961, 993, 1035, 1085, 1117, 1165, 1217, 1253, 1303
Offset: 1

Views

Author

Seiichi Manyama, May 08 2024

Keywords

Links

Crossrefs

Column k=10 of A372619.
Cf. A000010.

Programs

  • Mathematica
    Accumulate[Table[EulerPhi[10*n], {n, 1, 60}]]/4 (* Amiram Eldar, May 08 2024 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(10*k))/4;

Formula

a(n) ~ (25/(6*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

A372622 a(n) = (1/4) * Sum_{k=1..n} phi(5*k).

Original entry on oeis.org

1, 2, 4, 6, 11, 13, 19, 23, 29, 34, 44, 48, 60, 66, 76, 84, 100, 106, 124, 134, 146, 156, 178, 186, 211, 223, 241, 253, 281, 291, 321, 337, 357, 373, 403, 415, 451, 469, 493, 513, 553, 565, 607, 627, 657, 679, 725, 741, 783, 808, 840, 864, 916, 934, 984, 1008, 1044
Offset: 1

Views

Author

Seiichi Manyama, May 07 2024

Keywords

Links

Crossrefs

Partial sums of A359101.
Column k=5 of A372619.
Cf. A000010.

Programs

  • Mathematica
    Accumulate[Table[EulerPhi[5*n], {n, 1, 60}]]/4 (* Amiram Eldar, May 08 2024 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(5*k))/4;

Formula

a(n) ~ (25/(8*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

A372638 a(n) = (1/6) * Sum_{k=1..n} phi(7*k).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 19, 23, 29, 33, 43, 47, 59, 66, 74, 82, 98, 104, 122, 130, 144, 154, 176, 184, 204, 216, 234, 248, 276, 284, 314, 330, 350, 366, 394, 406, 442, 460, 484, 500, 540, 554, 596, 616, 640, 662, 708, 724, 773, 793, 825, 849, 901, 919, 959, 987, 1023
Offset: 1

Views

Author

Seiichi Manyama, May 08 2024

Keywords

Links

Crossrefs

Column k=7 of A372619.
Partial sums of A359102.
Cf. A000010.

Programs

  • Mathematica
    Accumulate[Table[EulerPhi[7*n], {n, 1, 60}]]/6 (* Amiram Eldar, May 08 2024 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(7*k))/6;

Formula

a(n) ~ (49/(16*Pi^2)) * n^2. - Amiram Eldar, May 08 2024
Showing 1-8 of 8 results.

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