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

1,2

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

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

Sequence in context: A363583 A355918 A003255 * A244365 A171014 A254050

Adjacent sequences: A263318 A263319 A263320 * A263322 A263323 A263324

Keyword

nonn

Author

Zhi-Wei Sun, Oct 14 2015

Status

approved