A237656 - OEIS

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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.

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

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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

Cf. A000290, A000720, A038107, A237643.

Sequence in context: A357467 A333009 A363229 * A371847 A097907 A198749

Adjacent sequences: A237653 A237654 A237655 * A237657 A237658 A237659

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 10 2014

STATUS

approved