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A237656
Least positive integer m such that {A000720(k^2): k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.
6
1, 5, 3, 6, 8, 10, 18, 17, 30, 41, 28, 43, 29, 33, 43, 27, 66, 47, 98, 105, 155, 114, 113, 100, 49, 62, 118, 146, 85, 125, 80, 117, 74, 101, 167, 137, 168, 282, 176, 287, 129, 178, 151, 140, 163, 139, 262, 267, 277, 234, 285, 188, 203, 163, 192, 239, 188, 241, 252, 252
OFFSET
1,2
COMMENTS
Conjecture: a(n) is always positive. Moreover, a(n) < 2*prime(n+1) - 2 for all n > 0.
Note that a(21) = 155 = 2*prime(22) - 3.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..1011 (n = 1..100 from Zhi-Wei Sun)
Zhi-Wei Sun, On a^n+bn modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.
EXAMPLE
a(5) = 8 since {A000720(k^2): k = 1, ..., 8} = {0, 2, 4, 6, 9, 11, 15, 18} contains a complete system of residues modulo 5, but {A000720(k^2): k = 1, ..., 7} contains no integer congruent to 3 modulo 5.
MATHEMATICA
q[m_, n_]:=Length[Union[Table[Mod[PrimePi[k^2], n], {k, 1, m}]]]
Do[Do[If[q[m, n]==n, Print[n, " ", m]; Goto[aa]], {m, n, 2*Prime[n+1]-3}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 10 2014
STATUS
approved