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

Author

Amiram Eldar, Mar 18 2021

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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]

PARI

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