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A234642 Smallest x such that x mod phi(x) = n, or 0 if no such x exists. 1

%I

%S 1,3,10,9,20,25,30,15,40,21,50,35,60,33,98,39,80,65,90,51,100,45,70,

%T 95,120,69,338,63,196,161,110,87,160,93,130,75,180,217,182,99,200,185,

%U 170,123,140,117,190,215,240,141,250,235,676,329,230,159,392,153,322

%N Smallest x such that x mod phi(x) = n, or 0 if no such x exists.

%C Conjecture: a(n) > 0 for all n. This would follow from a form of Goldbach's (binary) conjecture. Checked up to 10^7; largest term in that range is a(9972987) = 4178506411.

%C Pomerance proves that x = n (mod phi(x)) has at least two solutions for each n, but this allows x < n and so does not prove the conjecture above.

%C a(n) > 0 for all n <= 10^9. The largest term in that range is a(990429171) = 1050844225771. - _Donovan Johnson_, Feb 18 2014

%H Charles R Greathouse IV, <a href="/A234642/b234642.txt">Table of n, a(n) for n = 0..10000</a>

%H Carl Pomerance, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa26/aa2637.pdf">On the congruences σ(n) ≡ a (mod n) and n ≡ a (mod φ(n))</a>, Acta Arithmetica 26:3 (1974-1975), pp. 265-272.

%t A234642[n_]:=NestWhile[# + 1 &, 1, Not[Mod[#, EulerPhi[#]] == n] &] (* _JungHwan Min_, Dec 23 2015 *)

%t A234642[n_]:=Catch[Do[If[Mod[k, EulerPhi[k]] == n, Throw[k]], {k, Infinity}]] (* _JungHwan Min_, Dec 23 2015 *)

%t xmp[n_]:=Module[{x=1},While[Mod[x,EulerPhi[x]]!=n,x++];x]; Array[xmp,60,0] (* _Harvey P. Dale_, Jan 04 2016 *)

%o (PARI) a(n)=my(k=n);while(k++%eulerphi(k)!=n,);k

%Y Cf. A068494, A076495.

%K nonn,nice

%O 0,2

%A _Charles R Greathouse IV_, Dec 28 2013

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Last modified June 16 10:03 EDT 2019. Contains 324152 sequences. (Running on oeis4.)