A082849 - OEIS

%I #19 Mar 09 2021 04:06:43

%S 0,1,2,2,4,4,4,12,18,6,8,8,8,56,56,8,12,12,12,108,108,12,16,80,70,126,

%T 144,16,22,22,16,208,234,198,264,24,20,300,360,24,30,30,24,504,504,24,

%U 32,224,210,570,532,28,36,540,480,672,630,30,44,44,32,864,864,544,782

%N Product of cototient values of consecutive integers.

%H Harvey P. Dale, <a href="/A082849/b082849.txt">Table of n, a(n) for n = 1..1000</a>

%H Laurenţiu Panaitopol, <a href="https://www.jstor.org/stable/43678722">Asymptotical formula for a(n) = n - e(n)</a>, Bull. Math. Soc. Sci. Math. Roumanie, Vol. 42 (90), No. 3 (1999), pp. 271-277.

%F a(n) = A051953(n) * A051953(n+1).

%F Sum_{k=1..n} a(k) ~ c * n^3 + O(n^2 * log(n)^2), where c = (1/3) * (1 + Product_{p prime} (1 - 2/p^2)) - 4/Pi^2 = 0.03559329841.... (Panaitopol, 1999). - _Amiram Eldar_, Mar 09 2021

%t f[x_] := x-EulerPhi[x] tpr=Table[f[w+1]*f[w], {w, 1, 128}]

%t Times@@@Partition[Table[n-EulerPhi[n],{n,70}],2,1] (* _Harvey P. Dale_, Nov 17 2020 *)

%o (PARI) p=2;forprime(q=3,97,print1((p-eulerphi(p))*(q-eulerphi(q))", ");p=q) \\ _Charles R Greathouse IV_, Nov 16 2012

%Y Cf. A051953, A065474, A083538-A083555.

%K nonn

%O 1,3

%A _Labos Elemer_, May 22 2003