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

%I A342701 #16 Aug 28 2023 08:20:25
%S A342701 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 A342701 45,37,45,56,29,152,31,61,39,56,35,144,37,61,39,74,41,128,35,88,45,
%U A342701 161,47,192,49,82,51,74,95,216,43,97,75,203,59,304,91,88,63,122
%N A342701 a(n) is the second smallest k such that phi(n+k) = phi(k), or 0 if no such solution exists.
%C A342701 Sierpiński (1956) proved that there is at least one solution for all n>=1.
%C A342701 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 A342701 Schinzel (1958) observed that under the prime k-tuple conjecture there is a second solution for all even n.
%C A342701 Holt (2003) proved that there is a second solution for all even n <= 1.38 * 10^26595411.
%D A342701 Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, section B36, page 138-142.
%D A342701 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 217-219.
%D A342701 Wacław Sierpiński, Sur une propriété de la fonction phi(n), Publ. Math. Debrecen, Vol. 4 (1956), pp. 184-185.
%H A342701 Amiram Eldar, <a href="/A342701/b342701.txt">Table of n, a(n) for n = 1..10000</a>
%H A342701 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 A342701 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 A342701 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 A342701 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%e A342701 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 A342701 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 A342701 (PARI) a(n) = my(k=1, nb=0); while ((nb += (eulerphi(n+k)==eulerphi(k))) != 2, k++); k; \\ _Michel Marcus_, Mar 19 2021
%Y A342701 Cf. A000010, A001259, A001274, A007015, A217199.
%K A342701 nonn
%O A342701 1,1
%A A342701 _Amiram Eldar_, Mar 18 2021