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 1-3 of 3 results.

A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 24, 24, 24, 40, 40, 48, 72, 48, 64, 128, 96, 108, 144, 96, 120, 220, 176, 160, 240, 216, 216, 336, 224, 240, 480, 320, 320, 384, 288, 432, 648, 432, 384, 640, 480, 504, 840, 480, 528, 1012, 736, 672, 840, 640, 768, 1248, 936, 720, 960, 864, 1008
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Links

Crossrefs

Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).
Cf. A083481, A083539, A092517.

Programs

  • Haskell
    a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012
  • Maple
    a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 21 2022
  • Mathematica
    Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* Michael De Vlieger, Mar 25 2017 *)
Times@@@Partition[EulerPhi[Range[60]],2,1] (* Harvey P. Dale, Oct 29 2019 *)
  • PARI
    a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

Formula

a(n) = A000010(A002378(n)). - Amiram Eldar, Jul 10 2024
Sum_{k=1..n} a(k) = c * n^3 / 3 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024
a(n) = A058515(n)*A066813(n). - Amiram Eldar, May 07 2025

A065093 Convolution of A000010 with itself.

Original entry on oeis.org

1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120, 124, 188, 196, 276, 272, 404, 380, 544, 532, 716, 668, 968, 860, 1184, 1120, 1472, 1332, 1896, 1624, 2204, 2036, 2656, 2352, 3284, 2752, 3684, 3356, 4324, 3744, 5192, 4312, 5720, 5180, 6540, 5628, 7768, 6388, 8476
Offset: 1

Views

Author

Vladeta Jovovic, Nov 11 2001

Keywords

Links

Crossrefs

Cf. A000010, A000385, A055507, A065474, A330319.
Column k=2 of A340995.

Programs

  • Mathematica
    Table[Sum[EulerPhi[j]*EulerPhi[n-j], {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Aug 18 2021 *)
  • PARI
    { for (n=1, 1000, a=sum(k=1, n, eulerphi(k)*eulerphi(n+1-k)); write("b065093.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

Formula

a(n) = Sum_{k=1..n} phi(k)*phi(n+1-k), where phi is Euler totient function (A000010).
G.f.: (1/x)*(Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2)^2. - Ilya Gutkovskiy, Jan 31 2017
a(n) ~ (n^3/6) * c * Product_{primes p|n+1} ((p^3-2*p+1)/(p*(p^2-2))), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Ingham, 1927). - Amiram Eldar, Jul 13 2024

A335131 a(n) = Sum_{k=1..n} phi(k)*phi(k+1)*phi(k+2), where phi(k) = A000010(k) is Euler's totient function.

Original entry on oeis.org

2, 6, 22, 38, 86, 134, 278, 374, 614, 774, 1254, 1542, 2118, 2502, 3526, 4294, 6022, 6886, 8614, 9574, 12214, 13974, 17494, 19414, 23734, 26326, 32374, 35062, 41782, 45622, 55222, 60342, 68022, 72630, 82998, 90774, 106326, 113238, 128598, 136278, 156438, 166518
Offset: 1

Views

Author

Vaclav Kotesovec, May 24 2020

Keywords

Links

Crossrefs

Cf. A000010, A002088, A206256, A330319.

Programs

  • Mathematica
    Accumulate[Table[EulerPhi[k]*EulerPhi[k+1]*EulerPhi[k+2], {k, 1, 50}]]
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*eulerphi(k+1)*eulerphi(k+2)); \\ Michel Marcus, May 24 2020

Formula

a(n) ~ 3*c*n^4 / 8, where c = A206256 = Product_{p prime} (1 - 3/p^2) [Mirsky, 1949, p. 270, formula 30].
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.