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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A065642 A285109

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

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Last modified November 19 16:10 EST 2017. Contains 294936 sequences.