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 11-17 of 17 results.

A167347 Totally multiplicative sequence with a(p) = (p-1)*(p-3) = p^2-4p+3 for prime p.

Original entry on oeis.org

1, -1, 0, 1, 8, 0, 24, -1, 0, -8, 80, 0, 120, -24, 0, 1, 224, 0, 288, 8, 0, -80, 440, 0, 64, -120, 0, 24, 728, 0, 840, -1, 0, -224, 192, 0, 1224, -288, 0, -8, 1520, 0, 1680, 80, 0, -440, 2024, 0, 576, -64
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003958, A166589.

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]] - 3)^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-3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A003958(n) * A166589(n).
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) = 0.06874072991... . - Amiram Eldar, Dec 15 2022

A167352 Totally multiplicative sequence with a(p) = (p+1)*(p-3) = p^2-2p-3 for prime p.

Original entry on oeis.org

1, -3, 0, 9, 12, 0, 32, -27, 0, -36, 96, 0, 140, -96, 0, 81, 252, 0, 320, 108, 0, -288, 480, 0, 144, -420, 0, 288, 780, 0, 896, -243, 0, -756, 384, 0, 1292, -960, 0, -324, 1596, 0, 1760, 864, 0, -1440, 2112, 0, 1024, -432, 0, 1260, 2700, 0, 1152, -864, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003959, A166589.

Programs

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

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(3k) = 0 for k >= 1.
a(n) = A003959(n) * A166589(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 1/p^2 + 5/p^3 + 3/p^4) = 0.0629795941629... . - 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

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

A167361 Totally multiplicative sequence with a(p) = (p-3)^2 = p^2-6p+9 for prime p.

Original entry on oeis.org

1, 1, 0, 1, 4, 0, 16, 1, 0, 4, 64, 0, 100, 16, 0, 1, 196, 0, 256, 4, 0, 64, 400, 0, 16, 100, 0, 16, 676, 0, 784, 1, 0, 196, 64, 0, 1156, 256, 0, 4, 1444, 0, 1600, 64, 0, 400, 1936, 0, 256, 16, 0, 100, 2500, 0, 256, 16, 0, 676, 3136, 0, 3364, 784, 0, 1, 400
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]^2, {n, 1, 100}] (* G. C. Greubel, Jun 11 2016 *)

Formula

Multiplicative with a(p^e) = ((p-3)^2)^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-3)^2)^e(k).
a(3k) = 0 for k >= 1.
a(n) = A166589(n)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 5/p^2 - 3/p^3 - 9/p^4) = 0.07909568395... . - Amiram Eldar, Dec 15 2022

A167362 Totally multiplicative sequence with a(p) = (p-3)*(p+3) = p^2-9 for prime p.

Original entry on oeis.org

1, -5, 0, 25, 16, 0, 40, -125, 0, -80, 112, 0, 160, -200, 0, 625, 280, 0, 352, 400, 0, -560, 520, 0, 256, -800, 0, 1000, 832, 0, 952, -3125, 0, -1400, 640, 0, 1360, -1760, 0, -2000, 1672, 0, 1840, 2800, 0, -2600, 2200, 0, 1600, -1280, 0, 4000, 2800, 0, 1792
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A166589, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^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-3)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-3)*(p(k)+3))^e(k).
a(n) = A166589(n) * A166591(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 9/p^3 + 9/p^4) = 0.05980933853... . - 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 11-17 of 17 results.

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

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