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

%I A083542 #33 May 07 2025 03:03:43
%S A083542 1,2,4,8,8,12,24,24,24,40,40,48,72,48,64,128,96,108,144,96,120,220,
%T A083542 176,160,240,216,216,336,224,240,480,320,320,384,288,432,648,432,384,
%U A083542 640,480,504,840,480,528,1012,736,672,840,640,768,1248,936,720,960,864,1008
%N A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.
%H A083542 Reinhard Zumkeller, <a href="/A083542/b083542.txt">Table of n, a(n) for n = 1..10000</a>
%F A083542 a(n) = A000010(A002378(n)). - _Amiram Eldar_, Jul 10 2024
%F A083542 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
%F A083542 a(n) = A058515(n)*A066813(n). - _Amiram Eldar_, May 07 2025
%p A083542 a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
%p A083542 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jan 21 2022
%t A083542 Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* _Michael De Vlieger_, Mar 25 2017 *)
%t A083542 Times@@@Partition[EulerPhi[Range[60]],2,1] (* _Harvey P. Dale_, Oct 29 2019 *)
%o A083542 (Haskell)
%o A083542 a083542 n = a000010 n * a000010 (n + 1)
%o A083542 a083542_list = zipWith (*) (tail a000010_list) a000010_list
%o A083542 -- _Reinhard Zumkeller_, Apr 22 2012
%o A083542 (PARI) a(n) = eulerphi(n) * eulerphi(n+1); \\ _Amiram Eldar_, Jul 10 2024
%Y A083542 Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).
%Y A083542 Cf. A083481, A083539, A092517.
%K A083542 nonn,easy
%O A083542 1,2
%A A083542 _Labos Elemer_, May 21 2003