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 1-10 of 17 results. Next

A167312 Totally multiplicative sequence with a(p) = 2*(p-3) for prime p.

Original entry on oeis.org

1, -2, 0, 4, 4, 0, 8, -8, 0, -8, 16, 0, 20, -16, 0, 16, 28, 0, 32, 16, 0, -32, 40, 0, 16, -40, 0, 32, 52, 0, 56, -32, 0, -56, 32, 0, 68, -64, 0, -32, 76, 0, 80, 64, 0, -80, 88, 0, 64, -32, 0, 80, 100, 0, 64, -64, 0, -104, 112, 0, 116, -112, 0, 64, 80, 0, 128, 112
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A061142, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 08 2016 *)
f[p_, e_] := (2*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (2*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A061142(n) * A166589(n) = 2^bigomega(n) * A166589(n) = 2^A001222(n) * A166589(n).

A167313 Totally multiplicative sequence with a(p) = 3*(p-3) for prime p.

Original entry on oeis.org

1, -3, 0, 9, 6, 0, 12, -27, 0, -18, 24, 0, 30, -36, 0, 81, 42, 0, 48, 54, 0, -72, 60, 0, 36, -90, 0, 108, 78, 0, 84, -243, 0, -126, 72, 0, 102, -144, 0, -162, 114, 0, 120, 216, 0, -180, 132, 0, 144, -108, 0, 270, 150, 0, 144, -324, 0, -234, 168, 0, 174, -252, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 08 2016 *)
f[p_, e_] := (3*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (3*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165824(n) * A166589(n) = 3^bigomega(n) * A166589(n) = 3^A001222(n) * A166589(n).

A167314 Totally multiplicative sequence with a(p) = 4*(p-3) for prime p.

Original entry on oeis.org

1, -4, 0, 16, 8, 0, 16, -64, 0, -32, 32, 0, 40, -64, 0, 256, 56, 0, 64, 128, 0, -128, 80, 0, 64, -160, 0, 256, 104, 0, 112, -1024, 0, -224, 128, 0, 136, -256, 0, -512, 152, 0, 160, 512, 0, -320, 176, 0, 256, -256, 0, 640, 200, 0, 256, -1024, 0, -416, 224, 0, 232
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165825, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*4^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 08 2016 *)
f[p_, e_] := (4*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (4*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165825(n) * A166589(n) = 4^bigomega(n) * A166589(n) = 4^A001222(n) * A166589(n).

A167315 Totally multiplicative sequence with a(p) = 5*(p-3) for prime p.

Original entry on oeis.org

1, -5, 0, 25, 10, 0, 20, -125, 0, -50, 40, 0, 50, -100, 0, 625, 70, 0, 80, 250, 0, -200, 100, 0, 100, -250, 0, 500, 130, 0, 140, -3125, 0, -350, 200, 0, 170, -400, 0, -1250, 190, 0, 200, 1000, 0, -500, 220, 0, 400, -500, 0, 1250, 250, 0, 400, -2500, 0, -650, 280
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165826, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*5^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (5*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (5*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (5*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165826(n) * A166589(n) = 5^bigomega(n) * A166589(n) = 5^A001222(n) * A166589(n).

A167316 Totally multiplicative sequence with a(p) = 6*(p-3) for prime p.

Original entry on oeis.org

1, -6, 0, 36, 12, 0, 24, -216, 0, -72, 48, 0, 60, -144, 0, 1296, 84, 0, 96, 432, 0, -288, 120, 0, 144, -360, 0, 864, 156, 0, 168, -7776, 0, -504, 288, 0, 204, -576, 0, -2592, 228, 0, 240, 1728, 0, -720, 264, 0, 576, -864, 0, 2160, 300, 0, 576, -5184, 0, -936, 336
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165827, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*6^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (6*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (6*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (6*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165827(n) * A166589(n) = 6^bigomega(n) * A166589(n) = 6^A001222(n) * A166589(n).

A167317 Totally multiplicative sequence with a(p) = 7*(p-3) for prime p.

Original entry on oeis.org

1, -7, 0, 49, 14, 0, 28, -343, 0, -98, 56, 0, 70, -196, 0, 2401, 98, 0, 112, 686, 0, -392, 140, 0, 196, -490, 0, 1372, 182, 0, 196, -16807, 0, -686, 392, 0, 238, -784, 0, -4802, 266, 0, 280, 2744, 0, -980, 308, 0, 784, -1372, 0, 3430, 350, 0, 784, -9604, 0, -1274
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165828, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*7^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (7*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (7*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (7*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165828(n) * A166589(n) = 7^bigomega(n) * A166589(n) = 7^A001222(n) * A166589(n).

A167318 Totally multiplicative sequence with a(p) = 8*(p-3) for prime p.

Original entry on oeis.org

1, -8, 0, 64, 16, 0, 32, -512, 0, -128, 64, 0, 80, -256, 0, 4096, 112, 0, 128, 1024, 0, -512, 160, 0, 256, -640, 0, 2048, 208, 0, 224, -32768, 0, -896, 512, 0, 272, -1024, 0, -8192, 304, 0, 320, 4096, 0, -1280, 352, 0, 1024, -2048, 0, 5120, 400, 0, 1024, -16384
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (8*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (8*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165829(n) * A166589(n) = 8^bigomega(n) * A166589(n) = 8^A001222(n) * A166589(n).

A167319 Totally multiplicative sequence with a(p) = 9*(p-3) for prime p.

Original entry on oeis.org

1, -9, 0, 81, 18, 0, 36, -729, 0, -162, 72, 0, 90, -324, 0, 6561, 126, 0, 144, 1458, 0, -648, 180, 0, 324, -810, 0, 2916, 234, 0, 252, -59049, 0, -1134, 648, 0, 306, -1296, 0, -13122, 342, 0, 360, 5832, 0, -1620, 396, 0, 1296, -2916, 0, 7290, 450, 0, 1296, -26244
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165830, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*9^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (9*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (9*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (9*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165830(n) * A166589(n) = 9^bigomega(n) * A166589(n) = 9^A001222(n) * A166589(n).

A167320 Totally multiplicative sequence with a(p) = 10*(p-3) for prime p.

Original entry on oeis.org

1, -10, 0, 100, 20, 0, 40, -1000, 0, -200, 80, 0, 100, -400, 0, 10000, 140, 0, 160, 2000, 0, -800, 200, 0, 400, -1000, 0, 4000, 260, 0, 280, -100000, 0, -1400, 800, 0, 340, -1600, 0, -20000, 380, 0, 400, 8000, 0, -2000, 440, 0, 1600, -4000, 0, 10000, 500, 0, 1600
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165831, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*10^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (10*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (10*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (10*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165831(n) * A166589(n) = 10^bigomega(n) * A166589(n) = 10^A001222(n) * A166589(n).

A167341 Totally multiplicative sequence with a(p) = p*(p-3) = p^2-3p for prime p.

Original entry on oeis.org

1, -2, 0, 4, 10, 0, 28, -8, 0, -20, 88, 0, 130, -56, 0, 16, 238, 0, 304, 40, 0, -176, 460, 0, 100, -260, 0, 112, 754, 0, 868, -32, 0, -476, 280, 0, 1258, -608, 0, -80, 1558, 0, 1720, 352, 0, -920, 2068, 0, 784, -200
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)

Formula

Multiplicative with a(p^e) = (p*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = n * A166589(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 2/p^2 + 3/p^3) = 0.06544931251... . - Amiram Eldar, Dec 15 2022
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