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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 18 2021
STATUS
approved