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a(n) = smallest solution x to n + pi(x) = phi(x), or 0 if no such solution exists.
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%I #11 Nov 29 2024 21:34:20

%S 30,6,2,1,7,26,21,11,13,180,46,17,19,25,39,23,64,74,174,51,29,31,57,

%T 104,94,228,37,69,116,41,43,258,65,47,81,87,134,99,53,280,146,77,135,

%U 59,61,111,166,172,184,67,91,408,71,73,194,129,202,780,79,212,115,83

%N a(n) = smallest solution x to n + pi(x) = phi(x), or 0 if no such solution exists.

%C Although pi takes all values, solutions are not guaranteed since nontotient numbers exist.

%C First possible holes (k>10^6) are for n = -3, 63, 73, 144, 162, 185, 204, 228, 233, 256, 264,...

%C Using phi(x) >= x / (exp(gamma)*log(log(x)) + 2) and pi(x) <= x / log(x) all the holes > 0 are easily proved. - _Sean A. Irvine_, Nov 29 2024

%H Sean A. Irvine, <a href="/A073460/b073460.txt">Table of n, a(n) for n = -2..10000</a>

%F a(n)=Min{x; A000010(x)-A000720(x)=n}

%e n=13: Phi[13]=12, Pi[13]=6, a[12-6]=a(6)=13; n=63: no solution was found at x<=1000000.

%t Do[k = 1; While[ EulerPhi[k] != PrimePi[k] + n, k++ ]; Print[k], {n, -2, 62}]

%Y Cf. A000010, A000720, A037228, A037171, A073453-A073457, A072530, A072531, A073461, A073462.

%K nonn

%O -2,1

%A _Labos Elemer_, Aug 02 2002

%E Edited by _Robert G. Wilson v_, Aug 12 2002