A342701 a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.

3, 7, 5, 14, 9, 34, 7, 16, 15, 26, 11, 68, 39, 28, 15, 32, 33, 72, 25, 40, 35, 56, 17, 101, 45, 37, 45, 56, 29, 152, 31, 61, 39, 56, 35, 144, 37, 61, 39, 74, 41, 128, 35, 88, 45, 161, 47, 192, 49, 82, 51, 74, 95, 216, 43, 97, 75, 203, 59, 304, 91, 88, 63, 122

Offset

1,1

Comments

Sierpiński (1956) proved that there is at least one solution for all n>=1.

Schinzel (1958) proved that there are at least two solutions k to phi(n+k) = phi(k) for all n <= 8*10^47. Schinzel and Wakulicz (1959) increased this bound to 2*10^58.

Schinzel (1958) observed that under the prime k-tuple conjecture there is a second solution for all even n.

Holt (2003) proved that there is a second solution for all even n <= 1.38 * 10^26595411.

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, section B36, page 138-142.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 217-219.

Wacław Sierpiński, Sur une propriété de la fonction phi(n), Publ. Math. Debrecen, Vol. 4 (1956), pp. 184-185.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Jeffery J. Holt, The minimal number of solutions to phi(n)=phi(n+k), Math. Comp., Vol. 72, No. 244 (2003), pp. 2059-2061.

Andrzej Schinzel, Sur l'équation phi(x + k) = phi(x), Acta Arith., Vol. 4, No. 3 (1958), pp. 181-184.

Andrzej Schinzel and Andrzej Wakulicz, Sur l'équation phi(x+k)=phi(x). II, Acta Arith., Vol. 5, No. 4 (1959), pp. 425-426.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Example

a(1) = 3 since the solutions to the equation phi(1+k) = phi(k) are k = 1, 3, 15, 104, 164, ... (A001274), and 3 is the second solution.

Mathematica

f[n_, 0] = 0; f[n_, k0_] := Module[{k = f[n, k0 - 1] + 1}, While[EulerPhi[n + k] != EulerPhi[k], k++]; k]; Array[f[#, 2] &, 100]

Prog

(PARI) a(n) = my(k=1, nb=0); while ((nb += (eulerphi(n+k)==eulerphi(k))) != 2, k++); k; \\ Michel Marcus, Mar 19 2021

Crossrefs

Cf. A000010, A001259, A001274, A007015, A217199.

Sequence in context: A268261 A090940 A090916 * A184162 A369277 A324821

Adjacent sequences: A342698 A342699 A342700 * A342702 A342703 A342704

Keyword

nonn

Author

Amiram Eldar, Mar 18 2021

Status

approved