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

1, 3, 10, 9, 20, 25, 30, 15, 40, 21, 50, 35, 60, 33, 98, 39, 80, 65, 90, 51, 100, 45, 70, 95, 120, 69, 338, 63, 196, 161, 110, 87, 160, 93, 130, 75, 180, 217, 182, 99, 200, 185, 170, 123, 140, 117, 190, 215, 240, 141, 250, 235, 676, 329, 230, 159, 392, 153, 322

Offset: 0

Author

Charles R Greathouse IV, Dec 28 2013

Keywords

• nonn
• nice

Comments

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.

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.

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• Carl Pomerance, On the congruences σ(n) ≡ a (mod n) and n ≡ a (mod φ(n)), Acta Arithmetica 26:3 (1974-1975), pp. 265-272.

Crossrefs

Cf. A068494, A076495.

Programs

Mathematica

A234642[n_]:=NestWhile[# + 1 &, 1, Not[Mod[#, EulerPhi[#]] == n] &] (* JungHwan Min, Dec 23 2015 *)
A234642[n_]:=Catch[Do[If[Mod[k, EulerPhi[k]] == n, Throw[k]], {k, Infinity}]] (* JungHwan Min, Dec 23 2015 *)
xmp[n_]:=Module[{x=1},While[Mod[x,EulerPhi[x]]!=n,x++];x]; Array[xmp,60,0] (* Harvey P. Dale, Jan 04 2016 *)

PARI

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