A063748 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.

4, 9, 8, 25, 10, 49, 16, 27, 0, 121, 22, 169, 26, 55, 32, 289, 34, 361, 38, 85, 30, 529, 46, 133, 0, 187, 52, 841, 58, 961, 64, 253, 0, 323, 68, 1369, 74, 391, 76, 1681, 82, 1849, 86, 493, 70, 2209, 94, 589, 0, 667, 0, 2809, 106, 703, 104, 697, 0, 3481, 118, 3721, 122

Offset

2,1

Comments

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).

Links

T. D. Noe, Table of n, a(n) for n=2..1000

Formula

a(n)=Max{x : A051953(x)=n} if the inverse set is not empty; a(n)=0 if no inverse exists.

Example

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.

Mathematica

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 *)
Table[Module[{k = n^2}, While[And[k - EulerPhi@ k != n, k > 0], k--];
k], {n, 2, 62}] (* Michael De Vlieger, Mar 17 2017 *)

Crossrefs

Cf. A000010, A051953.

Cf. A063507 (least solution to x-phi(x)=n), A063740 (number of solutions to x-phi(x)=n).

Sequence in context: A289280 A077662 A063718 * A121920 A318279 A065642

Adjacent sequences: A063745 A063746 A063747 * A063749 A063750 A063751

Keyword

nonn

Author

Labos Elemer, Aug 13 2001

Extensions

Corrected and edited by T. D. Noe, Oct 30 2006

Status

approved