OFFSET
1,2
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.
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.
EXAMPLE
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.
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]]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 14 2015
STATUS
approved