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

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

Author

Labos Elemer, May 21 2003

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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