A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.

1, 3, 5, 7, 11, 13, 16, 19, 19, 19, 29, 29, 29, 37, 37, 59, 59, 59, 59, 59, 59, 59, 59, 101, 101, 101, 133, 133, 133, 133, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 175, 245, 269, 269, 269, 269, 379, 379

Offset: 1

Author

Zhi-Wei Sun, Oct 14 2015

Keywords

• nonn

Comments

Part (i) of the conjecture in A263319 implies that a(n) exists for any n > 0.

Conjecture: a(n) <= n^2 for all n > 0, and the only even term is a(7) = 16.

Examples

			a(7) = 16 since the 7 numbers phi(1)*pi(1^2) = 0, phi(2)*pi(2^2) = 2, phi(3)*pi(3^2) = 8, phi(4)*pi(4^2) = 12, phi(5)*pi(5^2) = 36, phi(6)*pi(6^2) = 22 and phi(7)*pi(7^2) = 90 are pairwise incongruent modulo 16, but not so modulo any positive integer smaller than 16.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.

Crossrefs

Cf. A000010, A000290, A000720, A038107, A263317, A263319.

Programs

Mathematica

f[n_]:=f[n]=EulerPhi[n]*PrimePi[n^2]
Le[n_,m_]:=Le[m,n]=Length[Union[Table[Mod[f[k],m],{k,1,n}]]]
Do[n=1;m=1;Label[aa];If[Le[n,m]==n,Goto[bb],m=m+1;Goto[aa]];
Label[bb];Print[n," ",m];If[n<50,n=n+1;Goto[aa]]]