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.

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A167308 Totally multiplicative sequence with a(p) = 7*(p+2) for prime p.

Original entry on oeis.org

1, 28, 35, 784, 49, 980, 63, 21952, 1225, 1372, 91, 27440, 105, 1764, 1715, 614656, 133, 34300, 147, 38416, 2205, 2548, 175, 768320, 2401, 2940, 42875, 49392, 217, 48020, 231, 17210368, 3185, 3724, 3087, 960400, 273, 4116, 3675, 1075648, 301, 61740, 315, 71344
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165828, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*7^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (7*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Formula

Multiplicative with a(p^e) = (7*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (7*(p(k)+2))^e(k).
a(n) = A165828(n) * A166590(n) = 7^bigomega(n) * A166590(n) = 7^A001222(n) * A166590(n).

A167309 Totally multiplicative sequence with a(p) = 8*(p+2) for prime p.

Original entry on oeis.org

1, 32, 40, 1024, 56, 1280, 72, 32768, 1600, 1792, 104, 40960, 120, 2304, 2240, 1048576, 152, 51200, 168, 57344, 2880, 3328, 200, 1310720, 3136, 3840, 64000, 73728, 248, 71680, 264, 33554432, 4160, 4864, 4032, 1638400, 312, 5376, 4800, 1835008, 344, 92160, 360
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (8*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Formula

Multiplicative with a(p^e) = (8*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)+2))^e(k).
a(n) = A165829(n) * A166590(n) = 8^bigomega(n) * A166590(n) = 8^A001222(n) * A166590(n).

A167310 Totally multiplicative sequence with a(p) = 9*(p+2) for prime p.

Original entry on oeis.org

1, 36, 45, 1296, 63, 1620, 81, 46656, 2025, 2268, 117, 58320, 135, 2916, 2835, 1679616, 171, 72900, 189, 81648, 3645, 4212, 225, 2099520, 3969, 4860, 91125, 104976, 279, 102060, 297, 60466176, 5265, 6156, 5103, 2624400, 351, 6804, 6075, 2939328, 387, 131220, 405
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165830, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*9^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (9*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Formula

Multiplicative with a(p^e) = (9*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (9*(p(k)+2))^e(k).
a(n) = A165830(n) * A166590(n) = 9^bigomega(n) * A166590(n) = 9^A001222(n) * A166590(n).

A167311 Totally multiplicative sequence with a(p) = 10*(p+2) for prime p.

Original entry on oeis.org

1, 40, 50, 1600, 70, 2000, 90, 64000, 2500, 2800, 130, 80000, 150, 3600, 3500, 2560000, 190, 100000, 210, 112000, 4500, 5200, 250, 3200000, 4900, 6000, 125000, 144000, 310, 140000, 330, 102400000, 6500, 7600, 6300, 4000000, 390, 8400, 7500
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165831, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*10^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (10*(p+2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

Formula

Multiplicative with a(p^e) = (10*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (10*(p(k)+2))^e(k).
a(n) = A165831(n) * A166590(n) = 10^bigomega(n) * A166590(n) = 10^A001222(n) * A166590(n).

A167340 Totally multiplicative sequence with a(p) = p*(p+2) = p^2+2p for prime p.

Original entry on oeis.org

1, 8, 15, 64, 35, 120, 63, 512, 225, 280, 143, 960, 195, 504, 525, 4096, 323, 1800, 399, 2240, 945, 1144, 575, 7680, 1225, 1560, 3375, 4032, 899, 4200, 1023, 32768, 2145, 2584, 2205, 14400, 1443, 3192, 2925, 17920, 1763, 7560, 1935, 9152, 7875, 4600
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Programs

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

Formula

Multiplicative with a(p^e) = (p*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+2))^e(k). a(n) = n * A166590(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 2*p - 1)) = 1.316691699195895375836915424544566393355705508235453271181975628362968836... - Vaclav Kotesovec, Sep 20 2020

A167346 Totally multiplicative sequence with a(p) = (p-1)*(p+2) = p^2+p-2 for prime p.

Original entry on oeis.org

1, 4, 10, 16, 28, 40, 54, 64, 100, 112, 130, 160, 180, 216, 280, 256, 304, 400, 378, 448, 540, 520, 550, 640, 784, 720, 1000, 864, 868, 1120, 990, 1024, 1300, 1216, 1512, 1600, 1404, 1512, 1800, 1792, 1720, 2160, 1890, 2080, 2800, 2200, 2254, 2560, 2916, 3136
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003958, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)
f[p_, e_] := ((p - 1)*(p + 2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2022 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, ((f[i,1]-1)*(f[i,1]+2))^f[i,2]); } \\ Amiram Eldar, Nov 05 2022

Formula

Multiplicative with a(p^e) = ((p-1)*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+2))^e(k).
a(n) = A003958(n) * A166590(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 3)) = 1.611922780552146990915794949248803526278171368254928942581015265238806543... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 2/p^4)) = 0.3809790887... . - Amiram Eldar, Nov 05 2022

A167355 Totally multiplicative sequence with a(p) = (p-2)*(p+2) = p^2-4 for prime p.

Original entry on oeis.org

1, 0, 5, 0, 21, 0, 45, 0, 25, 0, 117, 0, 165, 0, 105, 0, 285, 0, 357, 0, 225, 0, 525, 0, 441, 0, 125, 0, 837, 0, 957, 0, 585, 0, 945, 0, 1365, 0, 825, 0, 1677, 0, 1845, 0, 525, 0, 2205, 0, 2025, 0, 1425, 0, 2805, 0, 2457, 0, 1785, 0, 3477, 0, 3717, 0, 1125
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166586, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p-2)*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-2)*(p(k)+2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A166586(n) * A166590(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 4/p^3 + 4/p^4) = 0.128353657048... . - Amiram Eldar, Dec 15 2022

A167358 Totally multiplicative sequence with a(p) = (p+2)^2 = p^2+4p+4 for prime p.

Original entry on oeis.org

1, 16, 25, 256, 49, 400, 81, 4096, 625, 784, 169, 6400, 225, 1296, 1225, 65536, 361, 10000, 441, 12544, 2025, 2704, 625, 102400, 2401, 3600, 15625, 20736, 961, 19600, 1089, 1048576, 4225, 5776, 3969, 160000, 1521, 7056, 5625, 200704, 1849, 32400, 2025, 43264
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]^2, {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+2)^2)^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+2)^2)^e(k). a(n) = A166590(n)^2.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 4*p + 3)) = 1.1773018966974266400752906612246691227245078032189833736353235503076639420... - Vaclav Kotesovec, Sep 20 2020

A167359 Totally multiplicative sequence with a(p) = (p+2)*(p-3) = p^2-p-6 for prime p.

Original entry on oeis.org

1, -4, 0, 16, 14, 0, 36, -64, 0, -56, 104, 0, 150, -144, 0, 256, 266, 0, 336, 224, 0, -416, 500, 0, 196, -600, 0, 576, 806, 0, 924, -1024, 0, -1064, 504, 0, 1326, -1344, 0, -896, 1634, 0, 1800, 1664, 0, -2000, 2156, 0, 1296, -784, 0, 2400, 2750, 0, 1456
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166589, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+2)*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+2)*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A166590(n) * A166589(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 7/p^3 + 6/p^4) = 0.06114270465... . - Amiram Eldar, Dec 15 2022

A167360 Totally multiplicative sequence with a(p) = (p+2)*(p+3) = p^2+5p+6 for prime p.

Original entry on oeis.org

1, 20, 30, 400, 56, 600, 90, 8000, 900, 1120, 182, 12000, 240, 1800, 1680, 160000, 380, 18000, 462, 22400, 2700, 3640, 650, 240000, 3136, 4800, 27000, 36000, 992, 33600, 1122, 3200000, 5460, 7600, 5040, 360000, 1560, 9240, 7200, 448000, 1892, 54000, 2070
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166590, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p+2)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+2)*(p(k)+3))^e(k). a(n) = A166590(n) * A166591(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 5*p + 5)) = 1.1480407951783735490090642594369977652983537687209929674246821640934042061... - Vaclav Kotesovec, Sep 20 2020
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