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A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A117203 A081118 A003255 * A244365 A171014 A254050

Adjacent sequences:  A263318 A263319 A263320 * A263322 A263323 A263324

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 14 2015

STATUS

approved

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Last modified October 22 08:00 EDT 2019. Contains 328315 sequences. (Running on oeis4.)