A232862 -id:A232862 - cp's OEIS frontend

A232616 Least positive integer m such that {2^k - k: k = 1,...,m} contains a complete system of residues modulo n.

1, 2, 4, 5, 10, 6, 14, 10, 12, 18, 29, 13, 33, 22, 40, 19, 38, 18, 58, 21, 36, 58, 75, 26, 60, 66, 40, 64, 195, 53, 87, 36, 158, 67, 130, 37, 133, 94, 90, 42, 95, 42, 105, 112, 112, 140, 247, 51, 122, 94, 119, 120, 311, 54, 126, 90, 184, 223, 264, 61

Offset: 1

Zhi-Wei Sun, Nov 26 2013

nonn

By a result of the author (see arXiv:1312.1166), for any integers a and n > 0, the set {a^k - k: k = 1, ..., n^2} contains a complete system of residues modulo n. (We may also replace a^k - k by a^k + k.) Thus a(n) always exists and it does not exceed n^2.
Conjectures:
(i) a(n) < 2*(prime(n)-1) for all n > 0.
(ii) The Diophantine equation x^n - n = y^m with m, n, x, y > 1 only has two integral solutions: 2^5 - 5 = 3^3 and 2^7 - 7 = 11^2. Also, the Diophantine equation x^n + n = y^m with m, n, x, y > 1 only has two integral solutions: 5^2 + 2 = 3^3 and 5^3 + 3 = 2^7.

			a(3) = 4 since {2 - 1, 2^2 - 2, 2^3 - 3} = {1, 2, 5} does not contain a complete system of residues mod 3, but {2 - 1, 2^2 - 2, 2^3 - 3, 2^4 - 4} = {1, 2, 5, 12} does.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..700 from Zhi-Wei Sun)
Zhi-Wei Sun, On a^n + b*n modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000079, A000325, A231201, A231725, A232398, A232548, A232862.

L[m_,n_]:=Length[Union[Table[Mod[2^k-k,n],{k,1,m}]]]
Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2}];
Print[n," ",0];Label[aa];Continue,{n,1,60}]

A232894 Least positive integer m such that {Catalan(k) - k: k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

Original entry on oeis.org

1, 5, 4, 11, 16, 13, 31, 27, 18, 22, 34, 52, 45, 45, 31, 112, 57, 73, 113, 99, 64, 77, 114, 215, 134, 106, 89, 99, 127, 209, 161, 239, 135, 178, 96, 207, 185, 172, 157, 231, 174, 195, 309, 115, 274, 309, 386, 239, 200, 336, 188, 199, 181, 181, 116, 311, 229, 290, 663, 239

Offset: 1

Zhi-Wei Sun, Dec 02 2013

nonn

Conjecture: (i) Let n be any positive integer. Then 0 < a(n) <= n^2/2 + 7. Also, {Catalan(k) + k: k = 1, ..., [n^2/2] + 23} contains a complete system of residues modulo n, where [.] is the floor function.

(ii) For any integer n > 3, neither Catalan(n) - n nor Catalan(n) + n has the form x^m with m > 1 and x > 1.

			a(2) = 5 since Catalan(k) - k is even for each k = 1, 2, 3, 4, and Catalan(5) - 5 = 37 is odd.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..200 from Zhi-Wei Sun)

Cf. A000108, A232616, A232862, A232898.

L[m_,n_]:=Length[Union[Table[Mod[CatalanNumber[k]-k,n],{k,1,m}]]]
Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2/2+7}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,60}]

A232898 Least positive integer m such that {C(2k,k) + k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

Original entry on oeis.org

1, 2, 7, 5, 10, 12, 9, 24, 31, 22, 59, 25, 27, 30, 42, 56, 123, 66, 57, 72, 84, 78, 73, 132, 136, 57, 99, 80, 129, 211, 170, 226, 121, 170, 126, 129, 238, 218, 157, 132, 348, 198, 388, 103, 171, 166, 247, 181, 205, 352, 194, 136, 430, 226, 117, 224, 237, 292, 364, 241

Offset: 1

Zhi-Wei Sun, Dec 02 2013

nonn

Conjecture: (i) Let n be any positive integer. Then 0 < a(n) <= n^2/2 + 3. Also, {C(2k,k) - k: k = 1, ..., [n^2/2] + 15} contains a complete system of residues modulo n, where [.] is the floor function.

(ii) For any integer n > 2, neither C(2n,n) + n nor C(2n,n) - n has the form x^m with m > 1.

			a(2) = 2 since C(2*1,1) + 1 = 3 is odd and C(2*2,2) + 2 = 8 is even.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..250 from Zhi-Wei Sun)

Cf. A000984, A232616, A232862, A232894.

L[m_,n_]:=Length[Union[Table[Mod[Binomial[2k,k]+k,n],{k,1,m}]]]
Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2/2+3}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,60}]

A232891 Least positive integer m <= n^2/2 + 3 such that {k*prime(k): k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

Original entry on oeis.org

1, 3, 4, 11, 7, 7, 10, 17, 43, 13, 51, 22, 51, 36, 31, 49, 64, 71, 119, 73, 86, 68, 141, 110, 153, 85, 83, 86, 144, 81, 174, 127, 115, 87, 122, 138, 143, 134, 133, 142, 211, 229, 152, 104, 109, 177, 259, 142, 194, 176, 196, 311, 312, 193, 243, 197, 396, 169, 156, 171

Offset: 1

Zhi-Wei Sun, Dec 02 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) For any positive integer n not equal to 3, the number n*prime(n) + 1 cannot be a power x^m with m > 1.
(iii) There are infinitely many positive integers n with n - 1, n + 1, n + prime(n), n + prime(n)^2, n^2 + prime(n), n^2 + prime(n)^2 all prime.

			a(3) = 4 since 1*prime(1) = 2, 2*prime(2) === 3*prime(3) == 0 (mod 3), and 4*prime(4) = 28 == 1 (mod 3).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Zhi-Wei Sun)

Cf. A000040, A232616, A232861, A232862.

L[m_,n_]:=Length[Union[Table[Mod[k*Prime[k],n],{k,1,m}]]]
Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2/2+3}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,60}]

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A232616 Least positive integer m such that {2^k - k: k = 1,...,m} contains a complete system of residues modulo n.

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A232894 Least positive integer m such that {Catalan(k) - k: k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

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A232898 Least positive integer m such that {C(2k,k) + k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

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A232891 Least positive integer m <= n^2/2 + 3 such that {k*prime(k): k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

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