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

1,2

Links

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

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

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 *)

Prog

(Haskell)
a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012
(PARI) a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

Crossrefs

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

Cf. A083481, A083539, A092517.

Sequence in context: A123263 A008218 A073043 * A181533 A263981 A362936

Adjacent sequences: A083539 A083540 A083541 * A083543 A083544 A083545

Keyword

nonn,easy

Author

Labos Elemer, May 21 2003

Status

approved