A082849 Product of cototient values of consecutive integers.

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, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 300, 360, 24, 30, 30, 24, 504, 504, 24, 32, 224, 210, 570, 532, 28, 36, 540, 480, 672, 630, 30, 44, 44, 32, 864, 864, 544, 782

Offset: 1

Author

Labos Elemer, May 22 2003

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Laurenţiu Panaitopol, Asymptotical formula for a(n) = n - e(n), Bull. Math. Soc. Sci. Math. Roumanie, Vol. 42 (90), No. 3 (1999), pp. 271-277.

Crossrefs

Cf. A051953, A065474, A083538-A083555.

Programs

Mathematica

f[x_] := x-EulerPhi[x] tpr=Table[f[w+1]*f[w], {w, 1, 128}]
Times@@@Partition[Table[n-EulerPhi[n],{n,70}],2,1] (* Harvey P. Dale, Nov 17 2020 *)

PARI

p=2;forprime(q=3,97,print1((p-eulerphi(p))*(q-eulerphi(q))", ");p=q) \\ Charles R Greathouse IV, Nov 16 2012

Formula

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

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