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.

Previous Showing 11-20 of 20 results.

A167300 Totally multiplicative sequence with a(p) = 8*(p-2) for prime p.

Original entry on oeis.org

1, 0, 8, 0, 24, 0, 40, 0, 64, 0, 72, 0, 88, 0, 192, 0, 120, 0, 136, 0, 320, 0, 168, 0, 576, 0, 512, 0, 216, 0, 232, 0, 576, 0, 960, 0, 280, 0, 704, 0, 312, 0, 328, 0, 1536, 0, 360, 0, 1600, 0, 960, 0, 408, 0, 1728, 0, 1088, 0, 456, 0, 472, 0, 2560, 0, 2112, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166586.

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(2k) = 0 for k >= 1.
a(n) = A165829(n) * A166586(n) = 8^bigomega(n) * A166586(n) = 8^A001222(n) * A166586(n).

A167301 Totally multiplicative sequence with a(p) = 9*(p-2) for prime p.

Original entry on oeis.org

1, 0, 9, 0, 27, 0, 45, 0, 81, 0, 81, 0, 99, 0, 243, 0, 135, 0, 153, 0, 405, 0, 189, 0, 729, 0, 729, 0, 243, 0, 261, 0, 729, 0, 1215, 0, 315, 0, 891, 0, 351, 0, 369, 0, 2187, 0, 405, 0, 2025, 0, 1215, 0, 459, 0, 2187, 0, 1377, 0, 513, 0, 531, 0, 3645, 0, 2673, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165830, A166586.

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(2k) = 0 for k >= 1.
a(n) = A165830(n) * A166586(n) = 9^bigomega(n) * A166586(n) = 9^A001222(n) * A166586(n).

A167302 Totally multiplicative sequence with a(p) = 10*(p-2) for prime p.

Original entry on oeis.org

1, 0, 10, 0, 30, 0, 50, 0, 100, 0, 90, 0, 110, 0, 300, 0, 150, 0, 170, 0, 500, 0, 210, 0, 900, 0, 1000, 0, 270, 0, 290, 0, 900, 0, 1500, 0, 350, 0, 1100, 0, 390, 0, 410, 0, 3000, 0, 450, 0, 2500, 0, 1500, 0, 510, 0, 2700, 0, 1700, 0, 570, 0, 590, 0, 5000, 0, 3300
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165831, A166586.

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 19 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(2k) = 0 for k >= 1.
a(n) = A165831(n) * A166586(n) = 10^bigomega(n) * A166586(n) = 10^A001222(n) * A166586(n).

A167339 Totally multiplicative sequence with a(p) = p*(p-2) = p^2-2p for prime p.

Original entry on oeis.org

1, 0, 3, 0, 15, 0, 35, 0, 9, 0, 99, 0, 143, 0, 45, 0, 255, 0, 323, 0, 105, 0, 483, 0, 225, 0, 27, 0, 783, 0, 899, 0, 297, 0, 525, 0, 1295, 0, 429, 0, 1599, 0, 1763, 0, 135, 0, 2115, 0, 1225, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166586.

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 08 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(2k) = 0 for k >= 1.
a(n) = n * A166586(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 1/p^2 + 2/p^3) = 0.1016391193... . - Amiram Eldar, Dec 15 2022

A167345 Totally multiplicative sequence with a(p) = (p-1)*(p-2) = p^2-3p+2 for prime p.

Original entry on oeis.org

1, 0, 2, 0, 12, 0, 30, 0, 4, 0, 90, 0, 132, 0, 24, 0, 240, 0, 306, 0, 60, 0, 462, 0, 144, 0, 8, 0, 756, 0, 870, 0, 180, 0, 360, 0, 1260, 0, 264, 0, 1560, 0, 1722, 0, 48, 0, 2070, 0, 900, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003958, A166586.

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 *)

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(2k) = 0 for k >= 1.
a(n) = A003958(n) * A166586(n).
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) = 0.090842681006... . - Amiram Eldar, Dec 15 2022

A167354 Totally multiplicative sequence with a(p) = (p-2)^2 = p^2-4p+4 for prime p.

Original entry on oeis.org

1, 0, 1, 0, 9, 0, 25, 0, 1, 0, 81, 0, 121, 0, 9, 0, 225, 0, 289, 0, 25, 0, 441, 0, 81, 0, 1, 0, 729, 0, 841, 0, 81, 0, 225, 0, 1225, 0, 121, 0, 1521, 0, 1681, 0, 9, 0, 2025, 0, 625, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166586.

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(2k) = 0 for k >= 1.
a(n) = A166586(n)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 1 / (3 * Product_{p prime} (1 + 4/p^2)) = 0.08140990308... . - Amiram Eldar, Dec 15 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

A167356 Totally multiplicative sequence with a(p) = (p-2)*(p-3) = p^2-5p+6 for prime p.

Original entry on oeis.org

1, 0, 0, 0, 6, 0, 20, 0, 0, 0, 72, 0, 110, 0, 0, 0, 210, 0, 272, 0, 0, 0, 420, 0, 36, 0, 0, 0, 702, 0, 812, 0, 0, 0, 120, 0, 1190, 0, 0, 0, 1482, 0, 1640, 0, 0, 0, 1980, 0, 400, 0, 0, 0, 2550, 0, 432, 0, 0, 0, 3192, 0, 3422, 0, 0, 0, 660, 0, 4160, 0, 0, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166586, A166589.

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(2k) = 0 for k >= 1, a(3k) = 0 for k >= 1.
a(n) = A166586(n) * A166589(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 4/p^2 - 1/p^3 - 6/p^4) = 0.073139277512... . - Amiram Eldar, Dec 15 2022

A167357 Totally multiplicative sequence with a(p) = (p-2)*(p+3) = p^2+p-6 for prime p.

Original entry on oeis.org

1, 0, 6, 0, 24, 0, 50, 0, 36, 0, 126, 0, 176, 0, 144, 0, 300, 0, 374, 0, 300, 0, 546, 0, 576, 0, 216, 0, 864, 0, 986, 0, 756, 0, 1200, 0, 1400, 0, 1056, 0, 1716, 0, 1886, 0, 864, 0, 2250, 0, 2500, 0, 1800, 0, 2856, 0, 3024, 0, 2244, 0, 3534, 0, 3776, 0, 1800
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166586, 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(2k) = 0 for k >= 1.
a(n) = A166586(n) * A166591(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 2/p^2 + 5/p^3 + 6/p^4) = 0.1449357432... . - Amiram Eldar, Dec 15 2022

A359530 Multiplicative with a(p^e) = (p + 4)^e.

Original entry on oeis.org

1, 6, 7, 36, 9, 42, 11, 216, 49, 54, 15, 252, 17, 66, 63, 1296, 21, 294, 23, 324, 77, 90, 27, 1512, 81, 102, 343, 396, 33, 378, 35, 7776, 105, 126, 99, 1764, 41, 138, 119, 1944, 45, 462, 47, 540, 441, 162, 51, 9072, 121, 486, 147, 612, 57, 2058, 135, 2376, 161
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 26 2023

Keywords

Links

Crossrefs

Cf. A166589 (multiplicative with a(p^e) = (p-3)^e), A166586 (p-2), A003958 (p-1), A000027 (p), A003959 (p+1), A166590 (p+2), A166591 (p+3).

Programs

  • Mathematica
    g[p_, e_] := (p + 4)^e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-4*X))[n], ", "))
  • Python
    from math import prod
from sympy import factorint
def A359530(n): return prod((p+4)**e for p, e in factorint(n).items()) # Chai Wah Wu, Feb 26 2023

Formula

Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 4*p^(-s)).
Dirichlet g.f.: zeta(s-1) * (1 + 4/(2^s - 6)) * Product_{primes p, p>2} (1 + 4/(p^s - p - 4)).
Sum_{k=1..n} a(k) has an average value 2*c*zeta(r-1) * n^r / (3*log(6)), where r = 1 + log(3)/log(2) = 2.5849625007211561814537389439478165... and c = Product_{primes p, p>2} (1 + 4/(p^r - p - 4)) = 1.5747380964592139...
Previous Showing 11-20 of 20 results.

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