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

1,1

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).

References

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

Links

Table of n, a(n) for n=1..13.

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.

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.

Example

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

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 *)

Crossrefs

Cf. A000010, A001274.

Sequence in context: A213391 A328343 A309350 * A192798 A143609 A363430

Adjacent sequences: A300282 A300283 A300284 * A300286 A300287 A300288

Keyword

nonn,more

Author

Amiram Eldar, Mar 01 2018

Status

approved