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%I #12 Mar 18 2017 08:46:23
%S 4,9,8,25,10,49,16,27,0,121,22,169,26,55,32,289,34,361,38,85,30,529,
%T 46,133,0,187,52,841,58,961,64,253,0,323,68,1369,74,391,76,1681,82,
%U 1849,86,493,70,2209,94,589,0,667,0,2809,106,703,104,697,0,3481,118,3721,122
%N Greatest x that is a solution to x-phi(x)=n or zero if there is no solution, where phi(x) is Euler's totient function.
%C See A051953 for x-phi(x), the cototient function. Note that a(n)=0 for n in A005278. Also note that n=1 has an infinite number of solutions. If n is prime, then a(n)=n^2. If n is even, then a(n)<=2n. In particular, if n=p+1 for a prime p, then a(n)=2n-2. Also, if n=2^k, then a(n)=2n. If n>9 is odd and composite, then a(n)=pq, with p>q odd primes with p+q=n+1 and p-q minimal. We can take p=A078496((n+1)/2) and q=A078587((n+1)/2).
%H T. D. Noe, <a href="/A063748/b063748.txt">Table of n, a(n) for n=2..1000</a>
%F a(n)=Max{x : A051953(x)=n} if the inverse set is not empty; a(n)=0 if no inverse exists.
%e For n=15, the solutions are x=39 and x=55, so a(15)=55. Note that 55=5*11 and 5+11=n+1.
%t nn=10^4; lim=Floor[Sqrt[nn]]; mx=Table[0,{lim}]; Do[c=n-EulerPhi[n]; If[0<c<=lim, mx[[c]]=n], {n,nn}]; Rest[mx] (* _T. D. Noe_ *)
%t Table[Module[{k = n^2}, While[And[k - EulerPhi@ k != n, k > 0], k--];
%t k], {n, 2, 62}] (* _Michael De Vlieger_, Mar 17 2017 *)
%Y Cf. A000010, A051953.
%Y Cf. A063507 (least solution to x-phi(x)=n), A063740 (number of solutions to x-phi(x)=n).
%K nonn
%O 2,1
%A _Labos Elemer_, Aug 13 2001
%E Corrected and edited by _T. D. Noe_, Oct 30 2006
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