A300285 The number of solutions to phi(x) = phi(x+1) below 10^n, where phi(x) is the Euler totient function.

2, 3, 10, 17, 36, 68, 142, 306, 651, 1267, 2567, 5236, 10755

Offset: 1

Author

Amiram Eldar, Mar 01 2018

Keywords

• nonn
• more

Comments

Data extracted from A001274.

The terms were calculated by:

a(1)-a(2) - R. Ratat (1917).

a(3) - Victor L. Klee, Jr. (1947).

a(4)-a(5) - Mohan Lal and Paul Gillard (1972).

a(6) - David Ballew, Janell Case and Robert N. Higgins (1975).

a(7)-a(8) - Robert Baillie (1976).

a(9)-a(10) - Sidney West Graham, Jeffrey J. Holt, and Carl Pomerance (1999).

a(11) - T. D. Noe (2009).

a(12) - Jud McCranie (2012).

a(13) - Giovanni Resta (2014).

Examples

			Below 10^2 there are 3 solutions x = 1, 3, 15, hence a(2) = 3.
		

References

• R. Ratat, L'Intermédiaire des Mathématiciens, Vol. 24, pp. 101-102, 1917.

Links

• R. Baillie, Table of phi(n) = phi(n+1), Math. Comp., 30 (1976), pp. 189-190.
• David Ballew, Janell Case, and Robert N. Higgins, Table of phi(n)= phi(n+1), Math. Comput., Vol. 29, pp. 329-330, 1975.
• Sidney West Graham, Jeffrey J. Holt, and Carl Pomerance,On the solutions to phi(n)= phi(n+ k), Number Theory in Progress, Proceedings of the International Conference in Honor of the 60th Birthday of A. Schinzel, Poland, 1997, Walter de Gruyter, 1999, pp. 867-882.
• V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), p. 332.
• Mohan Lal and Paul Gillard, On the equation phi(n) = phi(n+k), Math. Comp., 26 (1972), pp. 579-583.
• Leo Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), pp. 22-23.

Crossrefs

Cf. A000010, A001274.

Programs

Mathematica

With[{s = Array[EulerPhi, 10^6]}, Array[Count[Range[10^# - 1], ?(s[[#]] == s[[# + 1]] &)] &, IntegerLength@ Length@ s - 1]] (* _Michael De Vlieger, Mar 04 2018 *)

Formula

According to Thomas Ordowski's conjecture in A001274, a(n) ~ 10^(C*n/3), where C = 9/Pi^2 = 0.911891... Numerically it seems that C ~ 0.93.