A057540 Birthday set of order 8: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7 and 8.

1, 41, 71, 169, 209, 239, 281, 391, 449, 559, 601, 631, 671, 769, 799, 839, 841, 881, 911, 1009, 1049, 1079, 1121, 1231, 1289, 1399, 1441, 1471, 1511, 1609, 1639, 1679, 1681, 1721, 1751, 1849, 1889, 1919, 1961, 2071, 2129, 2239, 2281, 2311, 2351, 2449

Offset: 1

Author

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

Keywords

• nonn
• easy

Examples

			2129 is on the list because it is congruent to 1 mod 2, -1 mod 3, 1 mod 4, -1 mod 5, -1 mod 6, 1 mod 7 and 1 mod 8.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)
• A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A007310, A057538, A057539 and A057541 are also birthday sets.

Programs

Mathematica

bso8Q[n_]:=Module[{s1=Mod[n,Range[2,8]],s2},s2=Abs[s1-Range[2,8]];AllTrue[ Thread[{s1,s2}],MemberQ[#,1]&]]; Select[Range[2500],bso8Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 18 2021 *)

PARI

Vec(x*(x^16 +40*x^15 +30*x^14 +98*x^13 +40*x^12 +30*x^11 +42*x^10 +110*x^9 +58*x^8 +110*x^7 +42*x^6 +30*x^5 +40*x^4 +98*x^3 +30*x^2 +40*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)) + O(x^100)) \\ Colin Barker, Mar 16 2015

Formula

G.f.: x*(x^16 +40*x^15 +30*x^14 +98*x^13 +40*x^12 +30*x^11 +42*x^10 +110*x^9 +58*x^8 +110*x^7 +42*x^6 +30*x^5 +40*x^4 +98*x^3 +30*x^2 +40*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)). - Colin Barker, Mar 16 2015

Extensions

Offset corrected to 1 by Ray Chandler, Jul 29 2019