A342701 - OEIS

%I #16 Aug 28 2023 08:20:25

%S 3,7,5,14,9,34,7,16,15,26,11,68,39,28,15,32,33,72,25,40,35,56,17,101,

%T 45,37,45,56,29,152,31,61,39,56,35,144,37,61,39,74,41,128,35,88,45,

%U 161,47,192,49,82,51,74,95,216,43,97,75,203,59,304,91,88,63,122

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

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

%C 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.

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

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

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

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

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

%H Amiram Eldar, <a href="/A342701/b342701.txt">Table of n, a(n) for n = 1..10000</a>

%H Jeffery J. Holt, <a href="http://dx.doi.org/10.1090/S0025-5718-03-01509-6">The minimal number of solutions to phi(n)=phi(n+k)</a>, Math. Comp., Vol. 72, No. 244 (2003), pp. 2059-2061.

%H Andrzej Schinzel, <a href="https://eudml.org/doc/206114">Sur l'équation phi(x + k) = phi(x)</a>, Acta Arith., Vol. 4, No. 3 (1958), pp. 181-184.

%H Andrzej Schinzel and Andrzej Wakulicz, <a href="https://eudml.org/doc/206421">Sur l'équation phi(x+k)=phi(x). II</a>, Acta Arith., Vol. 5, No. 4 (1959), pp. 425-426.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.

%e 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.

%t 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]

%o (PARI) a(n) = my(k=1, nb=0); while ((nb += (eulerphi(n+k)==eulerphi(k))) != 2, k++); k; \\ _Michel Marcus_, Mar 19 2021

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

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 18 2021