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

A109624 Totally multiplicative sequence with a(p) = (p-1)*(p+3) = p^2+2p-3 for prime p.

Original entry on oeis.org

1, 5, 12, 25, 32, 60, 60, 125, 144, 160, 140, 300, 192, 300, 384, 625, 320, 720, 396, 800, 720, 700, 572, 1500, 1024, 960, 1728, 1500, 896, 1920, 1020, 3125, 1680, 1600, 1920, 3600, 1440, 1980, 2304, 4000, 1760, 3600, 1932, 3500, 4608, 2860, 2300, 7500, 3600
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003958, A166591.

Programs

  • Mathematica
    f[p_, e_] := ((p - 1)*(p + 3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2022 *)
  • PARI
    a(n) = {f = factor(n); return (prod(k=1, #f~, ((f[k, 1]-1)*(f[k, 1]+3))^f[k, 2]));} \\ Michel Marcus, Jun 13 2013

Formula

Multiplicative with a(p^e) = ((p-1)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+3))^e(k).
a(n) = A003958(n) * A166591(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 2*p - 4)) = 1.471999388763656342016756485604184156984049961181587531678650682804811302... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 3/p^4)) = 0.6324191395... . - Amiram Eldar, Nov 05 2022

A167321 Totally multiplicative sequence with a(p) = 2*(p+3) for prime p.

Original entry on oeis.org

1, 10, 12, 100, 16, 120, 20, 1000, 144, 160, 28, 1200, 32, 200, 192, 10000, 40, 1440, 44, 1600, 240, 280, 52, 12000, 256, 320, 1728, 2000, 64, 1920, 68, 100000, 336, 400, 320, 14400, 80, 440, 384, 16000, 88, 2400, 92, 2800, 2304, 520, 100, 120000, 400, 2560
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A061142, A166591.

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 09 2016 *)
f[p_, e_] := (2*(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) = (2*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)+3))^e(k).
a(n) = A061142(n) * A166591(n) = 2^bigomega(n) * A166591(n) = 2^A001222(n) * A166591(n).

A167322 Totally multiplicative sequence with a(p) = 3*(p+3) for prime p.

Original entry on oeis.org

1, 15, 18, 225, 24, 270, 30, 3375, 324, 360, 42, 4050, 48, 450, 432, 50625, 60, 4860, 66, 5400, 540, 630, 78, 60750, 576, 720, 5832, 6750, 96, 6480, 102, 759375, 756, 900, 720, 72900, 120, 990, 864, 81000, 132, 8100, 138, 9450, 7776, 1170, 150, 911250, 900
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166591.

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 09 2016 *)
f[p_, e_] := (3*(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) = (3*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)+3))^e(k).
a(n) = A165824(n) * A166591(n) = 3^bigomega(n) * A166591(n) = 3^A001222(n) * A166591(n).

A167323 Totally multiplicative sequence with a(p) = 4*(p+3) for prime p.

Original entry on oeis.org

1, 20, 24, 400, 32, 480, 40, 8000, 576, 640, 56, 9600, 64, 800, 768, 160000, 80, 11520, 88, 12800, 960, 1120, 104, 192000, 1024, 1280, 13824, 16000, 128, 15360, 136, 3200000, 1344, 1600, 1280, 230400, 160, 1760, 1536, 256000, 176, 19200, 184, 22400
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165825, A166591.

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 09 2016 *)
f[p_, e_] := (4*(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) = (4*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)+3))^e(k).
a(n) = A165825(n) * A166591(n) = 4^bigomega(n) * A166591(n) = 4^A001222(n) * A166591(n).

A167324 Totally multiplicative sequence with a(p) = 5*(p+3) for prime p.

Original entry on oeis.org

1, 25, 30, 625, 40, 750, 50, 15625, 900, 1000, 70, 18750, 80, 1250, 1200, 390625, 100, 22500, 110, 25000, 1500, 1750, 130, 468750, 1600, 2000, 27000, 31250, 160, 30000, 170, 9765625, 2100, 2500, 2000, 562500, 200, 2750, 2400, 625000, 220, 37500, 230
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165826, A166591.

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(n) = A165826(n) * A166591(n) = 5^bigomega(n) * A166591(n) = 5^A001222(n) * A166591(n).

A167325 Totally multiplicative sequence with a(p) = 6*(p+3) for prime p.

Original entry on oeis.org

1, 30, 36, 900, 48, 1080, 60, 27000, 1296, 1440, 84, 32400, 96, 1800, 1728, 810000, 120, 38880, 132, 43200, 2160, 2520, 156, 972000, 2304, 2880, 46656, 54000, 192, 51840, 204, 24300000, 3024, 3600, 2880, 1166400, 240, 3960, 3456, 1296000, 264, 64800, 276, 75600
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165827, A166591.

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 21 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(n) = A165827(n) * A166591(n) = 6^bigomega(n) * A166591(n) = 6^A001222(n) * A166591(n).

A167326 Totally multiplicative sequence with a(p) = 7*(p+3) for prime p.

Original entry on oeis.org

1, 35, 42, 1225, 56, 1470, 70, 42875, 1764, 1960, 98, 51450, 112, 2450, 2352, 1500625, 140, 61740, 154, 68600, 2940, 3430, 182, 1800750, 3136, 3920, 74088, 85750, 224, 82320, 238, 52521875, 4116, 4900, 3920, 2160900, 280, 5390, 4704, 2401000, 308
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165828, A166591.

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 21 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(n) = A165828(n) * A166591(n) = 7^bigomega(n) * A166591(n) = 7^A001222(n) * A166591(n).

A167327 Totally multiplicative sequence with a(p) = 8*(p+3) for prime p.

Original entry on oeis.org

1, 40, 48, 1600, 64, 1920, 80, 64000, 2304, 2560, 112, 76800, 128, 3200, 3072, 2560000, 160, 92160, 176, 102400, 3840, 4480, 208, 3072000, 4096, 5120, 110592, 128000, 256, 122880, 272, 102400000, 5376, 6400, 5120, 3686400, 320, 7040, 6144, 4096000, 352, 153600
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166591.

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 21 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(n) = A165829(n) * A166591(n) = 8^bigomega(n) * A166591(n) = 8^A001222(n) * A166591(n).

A167328 Totally multiplicative sequence with a(p) = 9*(p+3) for prime p.

Original entry on oeis.org

1, 45, 54, 2025, 72, 2430, 90, 91125, 2916, 3240, 126, 109350, 144, 4050, 3888, 4100625, 180, 131220, 198, 145800, 4860, 5670, 234, 4920750, 5184, 6480, 157464, 182250, 288, 174960, 306, 184528125, 6804, 8100, 6480, 5904900, 360, 8910, 7776, 6561000, 396, 218700
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165830, A166591.

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 21 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(n) = A165830(n) * A166591(n) = 9^bigomega(n) * A166591(n) = 9^A001222(n) * A166591(n).

A167329 Totally multiplicative sequence with a(p) = 10*(p+3) for prime p.

Original entry on oeis.org

1, 50, 60, 2500, 80, 3000, 100, 125000, 3600, 4000, 140, 150000, 160, 5000, 4800, 6250000, 200, 180000, 220, 200000, 6000, 7000, 260, 7500000, 6400, 8000, 216000, 250000, 320, 240000, 340, 312500000, 8400, 10000, 8000, 9000000, 400, 11000, 9600, 10000000, 440, 300000
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165831, A166591.

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 21 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(n) = A165831(n) * A166591(n) = 10^bigomega(n) * A166591(n) = 10^A001222(n) * A166591(n).
Showing 1-10 of 17 results. Next

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

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